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Table of Contents Summary
PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Conventions
      1.3  Propositional calculus
      1.4  Other axiomatizations of classical propositional calculus
      1.5  Predicate calculus mostly without distinct variables
      1.6  Predicate calculus with distinct variables
      1.7  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 Tarksi-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  EXTENSIBLE 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
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  MISCELLANEA
      14.1  Definitional Examples
      14.2  Natural deduction examples
      14.3  Humor
      14.4  (Future - to be reviewed and classified)
PART 15  DEPRECATED SECTIONS
      15.1  Additional material on Group theory
      15.2  Additional material on Rings and Fields
      15.3  Complex vector spaces
      15.4  Normed complex vector spaces
      15.5  Operators on complex vector spaces
      15.6  Inner product (pre-Hilbert) spaces
      15.7  Complex Banach spaces
      15.8  Complex Hilbert spaces
      15.9  Hilbert Space Explorer
PART 16  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      16.1  Mathboxes for user contributions
      16.2  Mathbox for Stefan Allan
      16.3  Mathbox for Mario Carneiro
      16.4  Mathbox for Paul Chapman
      16.5  Mathbox for Drahflow
      16.6  Mathbox for Scott Fenton
      16.7  Mathbox for Anthony Hart
      16.8  Mathbox for Chen-Pang He
      16.9  Mathbox for Jeff Hoffman
      16.10  Mathbox for Wolf Lammen
      16.11  Mathbox for Frédéric Liné
      16.12  Mathbox for Jeff Hankins
      16.13  Mathbox for Jeff Madsen
      16.14  Mathbox for Rodolfo Medina
      16.15  Mathbox for Stefan O'Rear
      16.16  Mathbox for Steve Rodriguez
      16.17  Mathbox for Andrew Salmon
      16.18  Mathbox for Jarvin Udandy
      16.19  Mathbox for David A. Wheeler
      16.20  Mathbox for Alan Sare
      16.21  Mathbox for Jonathan Ben-Naim
      16.22  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  Conventions
      1.3  Propositional calculus
            1.3.1  Recursively define primitive wffs for propositional calculus   wn 5
            1.3.2  The axioms of propositional calculus   ax-1 7
            1.3.3  Logical implication   mp2b 11
            1.3.4  Logical negation   con4d 99
            1.3.5  Logical equivalence   wb 178
            1.3.6  Logical disjunction and conjunction   wo 359
            1.3.7  Miscellaneous theorems of propositional calculus   pm5.21nd 873
            1.3.8  Abbreviated conjunction and disjunction of three wff's   w3o 938
            1.3.9  Logical 'nand' (Sheffer stroke)   wnan 1292
            1.3.10  Logical 'xor'   wxo 1300
            1.3.11  True and false constants   wtru 1312
            1.3.12  Truth tables   truantru 1332
            1.3.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1358
            1.3.14  Half-adders and full adders in propositional calculus   whad 1374
      1.4  Other axiomatizations of classical propositional calculus
            1.4.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1400
            1.4.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1418
            1.4.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1429
            1.4.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1435
            1.4.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1454
            1.4.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1458
            1.4.7  Deriving the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1473
            1.4.8  Deriving the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1496
            1.4.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1509
            1.4.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1528
      1.5  Predicate calculus mostly without distinct variables
            1.5.1  "Pure" (equality-free) predicate calculus axioms ax-5, ax-6, ax-7, ax-gen   wal 1532
            1.5.2  Introduce equality axioms ax-8, ax-11, ax-13, and ax-14   cv 1618
            1.5.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1628
            1.5.4  Introduce equality axioms ax-9v and ax-12   ax-9v 1632
            1.5.5  Derive ax-12o from ax-12   ax12o10lem1 1635
            1.5.6  Derive ax-10   ax10lem16 1665
            1.5.7  Derive ax-9 from the weaker version ax-9v   ax9 1683
            1.5.8  Introduce Axiom of Existence ax-9   ax-9 1684
            1.5.9  Derive ax-4, ax-5o, and ax-6o   ax4 1691
            1.5.10  "Pure" predicate calculus including ax-4, without distinct variables   a4i 1699
            1.5.11  Equality theorems without distinct variables   ax9o 1814
            1.5.12  Axioms ax-10 and ax-11   ax10o 1834
            1.5.13  Substitution (without distinct variables)   wsb 1882
            1.5.14  Theorems using axiom ax-11   equs5a 1911
      1.6  Predicate calculus with distinct variables
            1.6.1  Derive the axiom of distinct variables ax-16   a4imv 1922
            1.6.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1939
            1.6.3  Theorems without distinct variables that use axiom ax-11o   ax11b 1942
            1.6.4  Predicate calculus with distinct variables (cont.)   ax11v 1990
            1.6.5  More substitution theorems   equsb3lem 2061
            1.6.6  Existential uniqueness   weu 2114
      1.7  Other axiomatizations related to classical predicate calculus
            1.7.1  Predicate calculus with all distinct variables   ax-7d 2204
            1.7.2  Aristotelian logic: Assertic syllogisms   barbara 2210
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 2234
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2239
            2.1.3  Class form not-free predicate   wnfc 2372
            2.1.4  Negated equality and membership   wne 2412
            2.1.5  Restricted quantification   wral 2509
            2.1.6  The universal class   cvv 2727
            2.1.7  Conditional equality (experimental)   wcdeq 2904
            2.1.8  Russell's Paradox   ru 2920
            2.1.9  Proper substitution of classes for sets   wsbc 2921
            2.1.10  Proper substitution of classes for sets into classes   csb 3009
            2.1.11  Define basic set operations and relations   cdif 3075
            2.1.12  Subclasses and subsets   df-ss 3089
            2.1.13  The difference, union, and intersection of two classes   difeq1 3204
            2.1.14  The empty set   c0 3362
            2.1.15  "Weak deduction theorem" for set theory   cif 3470
            2.1.16  Power classes   cpw 3530
            2.1.17  Unordered and ordered pairs   csn 3544
            2.1.18  The union of a class   cuni 3727
            2.1.19  The intersection of a class   cint 3760
            2.1.20  Indexed union and intersection   ciun 3803
            2.1.21  Disjointness   wdisj 3891
            2.1.22  Binary relations   wbr 3920
            2.1.23  Ordered-pair class abstractions (class builders)   copab 3973
            2.1.24  Transitive classes   wtr 4010
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4028
            2.2.2  Derive the Axiom of Separation   axsep 4037
            2.2.3  Derive the Null Set Axiom   zfnuleu 4043
            2.2.4  Theorems requiring subset and intersection existence   nalset 4048
            2.2.5  Theorems requiring empty set existence   class2set 4072
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4082
            2.3.2  Derive the Axiom of Pairing   zfpair 4106
            2.3.3  Ordered pair theorem   opnz 4135
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4164
            2.3.5  Power class of union and intersection   pwin 4190
            2.3.6  Epsilon and identity relations   cep 4196
            2.3.7  Partial and complete ordering   wpo 4205
            2.3.8  Founded and well-ordering relations   wfr 4242
            2.3.9  Ordinals   word 4284
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4403
            2.4.2  Ordinals (continued)   ordon 4465
            2.4.3  Transfinite induction   tfi 4535
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4547
            2.4.5  Peano's postulates   peano1 4566
            2.4.6  Finite induction (for finite ordinals)   find 4572
            2.4.7  Functions and relations   cxp 4578
            2.4.8  Operations   co 5710
            2.4.9  "Maps to" notation   elmpt2cl 5913
            2.4.10  Function operation   cof 5928
            2.4.11  First and second members of an ordered pair   c1st 5972
            2.4.12  Function transposition   ctpos 6085
            2.4.13  Curry and uncurry   ccur 6124
            2.4.14  Proper subset relation   crpss 6128
            2.4.15  Definite description binder (inverted iota)   cio 6141
            2.4.16  Cantor's Theorem   canth 6178
            2.4.17  Undefined values and restricted iota (description binder)   cund 6180
            2.4.18  Functions on ordinals; strictly monotone ordinal functions   iunon 6241
            2.4.19  "Strong" transfinite recursion   crecs 6273
            2.4.20  Recursive definition generator   crdg 6308
            2.4.21  Finite recursion   frfnom 6333
            2.4.22  Abian's "most fundamental" fixed point theorem   abianfplem 6356
            2.4.23  Ordinal arithmetic   c1o 6358
            2.4.24  Natural number arithmetic   nna0 6488
            2.4.25  Equivalence relations and classes   wer 6543
            2.4.26  The mapping operation   cmap 6658
            2.4.27  Infinite Cartesian products   cixp 6703
            2.4.28  Equinumerosity   cen 6746
            2.4.29  Schroeder-Bernstein Theorem   sbthlem1 6856
            2.4.30  Equinumerosity (cont.)   xpf1o 6908
            2.4.31  Pigeonhole Principle   phplem1 6925
            2.4.32  Finite sets   onomeneq 6935
            2.4.33  Finite intersections   cfi 7048
            2.4.34  Hall's marriage theorem   marypha1lem 7070
            2.4.35  Supremum   csup 7077
            2.4.36  Ordinal isomorphism, Hartog's theorem   coi 7108
            2.4.37  Hartogs function, order types, weak dominance   char 7154
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7190
            2.5.2  Axiom of Infinity equivalents   inf0 7206
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7223
            2.6.2  Existence of omega (the set of natural numbers)   omex 7228
            2.6.3  Cantor normal form   ccnf 7246
            2.6.4  Transitive closure   trcl 7294
            2.6.5  Rank   cr1 7318
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7439
            2.6.7  Cardinal numbers   ccrd 7452
            2.6.8  Axiom of Choice equivalents   wac 7626
            2.6.9  Cardinal number arithmetic   ccda 7677
            2.6.10  The Ackermann bijection   ackbij2lem1 7729
            2.6.11  Cofinality (without Axiom of Choice)   cflem 7756
            2.6.12  Eight inequivalent definitions of finite set   sornom 7787
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 7926
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 7969
            3.2.2  AC equivalents: well ordering, Zorn's lemma   numthcor 8005
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8052
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8076
            3.2.5  Cofinality using Axiom of Choice   alephreg 8084
      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 8184
            4.1.2  Weak universes   cwun 8202
            4.1.3  Tarski's classes   ctsk 8250
            4.1.4  Grothendieck's universes   cgru 8292
      4.2  ZFC Set Theory plus the Tarksi-Grothendieck Axiom
            4.2.1  Introduce the Tarksi-Grothendieck Axiom   ax-groth 8325
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8328
            4.2.3  Tarski map function   ctskm 8339
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 8346
            5.1.2  Final derivation of real and complex number postulates   axaddf 8647
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8673
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 8698
            5.2.2  Infinity and the extended real number system   cpnf 8744
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 8774
            5.2.4  Ordering on reals   lttr 8779
            5.2.5  Initial properties of the complex numbers   mul12 8858
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 8905
            5.3.2  Subtraction   cmin 8917
            5.3.3  Multiplication   muladd 9092
            5.3.4  Ordering on reals (cont.)   gt0ne0 9119
            5.3.5  Reciprocals   ixi 9277
            5.3.6  Division   cdiv 9303
            5.3.7  Ordering on reals (cont.)   elimgt0 9472
            5.3.8  Completeness Axiom and Suprema   fimaxre 9581
            5.3.9  Imaginary and complex number properties   inelr 9616
            5.3.10  Function operation analogue theorems   ofsubeq0 9623
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9626
            5.4.2  Principle of mathematical induction   nnind 9644
            5.4.3  Decimal representation of numbers   c2 9675
            5.4.4  Some properties of specific numbers   0p1e1 9719
            5.4.5  The Archimedean property   nnunb 9840
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 9844
            5.4.7  Integers (as a subset of complex numbers)   cz 9903
            5.4.8  Decimal arithmetic   cdc 10003
            5.4.9  Upper partititions of integers   cuz 10109
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10190
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10195
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10221
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10233
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10328
            5.5.3  Supremum on the extended reals   xrsupexmnf 10501
            5.5.4  Real number intervals   cioo 10534
            5.5.5  Finite intervals of integers   cfz 10660
            5.5.6  Half-open integer ranges   cfzo 10748
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 10802
            5.6.2  The modulo (remainder) operation   cmo 10851
            5.6.3  The infinite sequence builder "seq"   om2uz0i 10888
            5.6.4  Integer powers   cexp 10982
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11160
            5.6.6  Factorial function   cfa 11166
            5.6.7  The binomial coefficient operation   cbc 11193
            5.6.8  The ` # ` (finite set size) function   chash 11215
            5.6.9  Words over a set   cword 11280
            5.6.10  Longer string literals   cs2 11368
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11438
            5.7.2  Real and imaginary parts; conjugate   ccj 11458
            5.7.3  Square root; absolute value   csqr 11595
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 11821
            5.8.2  Limits   cli 11835
            5.8.3  Finite and infinite sums   csu 12035
            5.8.4  The binomial theorem   binomlem 12164
            5.8.5  Infinite sums (cont.)   isumshft 12172
            5.8.6  Miscellaneous converging and diverging sequences   divrcnv 12185
            5.8.7  Arithmetic series   arisum 12192
            5.8.8  Geometric series   expcnv 12196
            5.8.9  Ratio test for infinite series convergence   cvgrat 12213
            5.8.10  Mertens' theorem   mertenslem1 12214
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12217
            5.9.2  _e is irrational   eirrlem 12356
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12361
            5.10.2  The reals are uncountable   rpnnen2lem1 12367
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12400
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12403
            6.1.3  The divides relation   cdivides 12405
            6.1.4  The division algorithm   divalglem0 12466
            6.1.5  Bit sequences   cbits 12484
            6.1.6  The greatest common divisor operator   cgcd 12559
            6.1.7  Bézout's identity   bezoutlem1 12591
            6.1.8  Algorithms   nn0seqcvgd 12614
            6.1.9  Euclid's Algorithm   eucalgval2 12625
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 12632
            6.2.2  Properties of the canonical representation of a rational   cnumer 12678
            6.2.3  Euler's theorem   codz 12705
            6.2.4  Pythagorean Triples   coprimeprodsq 12736
            6.2.5  The prime count function   cpc 12763
            6.2.6  Pocklington's theorem   prmpwdvds 12825
            6.2.7  Infinite primes theorem   unbenlem 12829
            6.2.8  Sum of prime reciprocals   prmreclem1 12837
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 12844
            6.2.10  Lagrange's four-square theorem   cgz 12850
            6.2.11  Van der Waerden's theorem   cvdwa 12886
            6.2.12  Ramsey's theorem   cram 12920
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 12952
            6.2.14  Specific prime numbers   4nprm 12980
            6.2.15  Very large primes   1259lem1 13003
PART 7  EXTENSIBLE STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13018
            7.1.2  Slot definitions   cplusg 13082
            7.1.3  Definition of the structure product   crest 13199
            7.1.4  Definition of the structure quotient   cordt 13272
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13380
            7.2.2  Algebraic closure systems   isacs 13398
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13410
            8.1.2  Opposite category   coppc 13458
            8.1.3  Monomorphisms and epimorphisms   cmon 13475
            8.1.4  Sections, inverses, isomorphisms   csect 13491
            8.1.5  Subcategories   cssc 13528
            8.1.6  Functors   cfunc 13572
            8.1.7  Full & faithful functors   cful 13620
            8.1.8  Natural transformations and the functor category   cnat 13659
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 13729
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 13751
            8.3.2  The category of categories   ccatc 13770
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 13786
            8.4.2  Functor evaluation   cevlf 13827
            8.4.3  Hom functor   chof 13866
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 13918
            9.2.2  Lattices   clat 13995
            9.2.3  The dual of an ordered set   codu 14076
            9.2.4  Subset order structures   cipo 14098
            9.2.5  Distributive lattices   latmass 14126
            9.2.6  Posets and lattices as relations   cps 14136
            9.2.7  Directed sets, nets   cdir 14185
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14196
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14248
            10.1.3  Ordered group sum operation   gsumvallem1 14283
            10.1.4  Free monoids   cfrmd 14304
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14324
            10.2.2  Subgroups and Quotient groups   csubg 14450
            10.2.3  Elementary theory of group homomorphisms   cghm 14515
            10.2.4  Isomorphisms of groups   cgim 14556
            10.2.5  Group actions   cga 14578
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 14604
            10.2.7  Centralizers and centers   ccntz 14626
            10.2.8  The opposite group   coppg 14653
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 14675
            10.2.10  Direct products   clsm 14780
            10.2.11  Free groups   cefg 14850
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 14924
            10.3.2  Cyclic groups   ccyg 14999
            10.3.3  Group sum operation   gsumval3a 15024
            10.3.4  Internal direct products   cdprd 15066
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15135
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15160
            10.4.2  Definition and basic properties   crg 15172
            10.4.3  Opposite ring   coppr 15239
            10.4.4  Divisibility   cdsr 15255
            10.4.5  Ring homomorphisms   crh 15329
      10.5  Division rings and Fields
            10.5.1  Definition and basic properties   cdr 15347
            10.5.2  Subrings of a ring   csubrg 15376
            10.5.3  Absolute value (abstract algebra)   cabv 15416
            10.5.4  Star rings   cstf 15443
      10.6  Left Modules
            10.6.1  Definition and basic properties   clmod 15462
            10.6.2  Subspaces and spans in a left module   clss 15524
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 15611
            10.6.4  Subspace sum; bases for a left module   clbs 15662
      10.7  Vector Spaces
            10.7.1  Definition and basic properties   clvec 15690
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 15753
            10.8.2  Two-sided ideals and quotient rings   c2idl 15815
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 15825
            10.8.4  Nonzero rings   cnzr 15841
            10.8.5  Left regular elements. More kinds of ring   crlreg 15852
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 15882
      10.10  Abstract Multivariate Polynomials
            10.10.1  Definition and basic properties   cmps 15919
            10.10.2  Polynomial evaluation   evlslem4 16077
            10.10.3  Univariate Polynomials   cps1 16082
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16201
            10.11.2  Algebraic constructions based on the complexes   czrh 16283
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16360
            10.12.2  Orthocomplements and closed subspaces   cocv 16392
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16432
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16463
            11.1.2  TopBases for topologies   isbasisg 16517
            11.1.3  Examples of topologies   distop 16565
            11.1.4  Closure and interior   ccld 16585
            11.1.5  Neighborhoods   cnei 16666
            11.1.6  Limit points and perfect sets   clp 16698
            11.1.7  Subspace topologies   restrcl 16720
            11.1.8  Order topology   ordtbaslem 16750
            11.1.9  Limits and Continuity in topological spaces   ccn 16786
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 16866
            11.1.11  Compactness   ccmp 16945
            11.1.12  Connectedness   ccon 16969
            11.1.13  First- and second-countability   c1stc 16995
            11.1.14  Local topological properties   clly 17022
            11.1.15  Compactly generated spaces   ckgen 17060
            11.1.16  Product topologies   ctx 17087
            11.1.17  Continuous function-builders   cnmptid 17187
            11.1.18  Quotient maps and quotient topology   ckq 17216
            11.1.19  Homeomorphisms   chmeo 17276
      11.2  Filters and filter bases
            11.2.1  Filter Bases   cfbas 17350
            11.2.2  Filters   cfil 17372
            11.2.3  Ultrafilters   cufil 17426
            11.2.4  Filter limits   cfm 17460
            11.2.5  Topological groups   ctmd 17585
            11.2.6  Infinite group sum on topological groups   ctsu 17640
            11.2.7  Topological rings, fields, vector spaces   ctrg 17670
      11.3  Metric spaces
            11.3.1  Basic metric space properties   cxme 17714
            11.3.2  Metric space balls   blfval 17779
            11.3.3  Open sets of a metric space   mopnval 17816
            11.3.4  Continuity in metric spaces   metcnp3 17918
            11.3.5  Examples of metric spaces   dscmet 17927
            11.3.6  Normed algebraic structures   cnm 17931
            11.3.7  Normed space homomorphisms (bounded linear operators)   cnmo 18046
            11.3.8  Topology on the Reals   qtopbaslem 18099
            11.3.9  Topological definitions using the reals   cii 18211
            11.3.10  Path homotopy   chtpy 18297
            11.3.11  The fundamental group   cpco 18330
            11.3.12  Complex left modules   cclm 18392
            11.3.13  Complex pre-Hilbert space   ccph 18434
            11.3.14  Convergence and completeness   ccfil 18510
            11.3.15  Baire's Category Theorem   bcthlem1 18578
            11.3.16  Banach spaces and complex Hilbert spaces   ccms 18586
            11.3.17  Minimizing Vector Theorem   minveclem1 18620
            11.3.18  Projection theorem   pjthlem1 18633
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 18640
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 18654
            12.2.2  Lebesgue integration   cmbf 18801
      12.3  Derivatives
            12.3.1  Real and Complex Differentiation   climc 19044
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19229
            13.1.2  Polynomial degrees   cmdg 19271
            13.1.3  The division algorithm for univariate polynomials   cmn1 19343
            13.1.4  Elementary properties of complex polynomials   cply 19398
            13.1.5  The Division algorithm for polynomials   cquot 19502
            13.1.6  Algebraic numbers   caa 19526
            13.1.7  Liouville's approximation theorem   aalioulem1 19544
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 19564
            13.2.2  Uniform convergence   culm 19587
            13.2.3  Power series   pserval 19618
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 19651
            13.3.2  Properties of pi = 3.14159...   pilem1 19659
            13.3.3  Mapping of the exponential function   efgh 19735
            13.3.4  The natural logarithm on complex numbers   clog 19744
            13.3.5  Solutions of quardatic, cubic, and quartic equations   quad2 19967
            13.3.6  Inverse trigonometric functions   casin 19990
            13.3.7  The Birthday Problem   log2ublem1 20074
            13.3.8  Areas in R^2   carea 20082
            13.3.9  More miscellaneous converging sequences   rlimcnp 20092
            13.3.10  Inequality of arithmetic and geometric means   cvxcl 20111
            13.3.11  Euler-Mascheroni constant   cem 20118
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20138
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20142
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20150
            13.4.4  Number-theoretical functions   ccht 20160
            13.4.5  Perfect Number Theorem   mersenne 20298
            13.4.6  Characters of Z/nZ   cdchr 20303
            13.4.7  Bertrand's postulate   bcctr 20346
            13.4.8  Legendre symbol   clgs 20365
            13.4.9  Quadratic Reciprocity   lgseisenlem1 20420
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 20434
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 20450
            13.4.12  The Prime Number Theorem   mudivsum 20511
            13.4.13  Ostrowski's theorem   abvcxp 20596
PART 14  MISCELLANEA
      14.1  Definitional Examples
      14.2  Natural deduction examples
      14.3  Humor
            14.3.1  April Fool's theorem   avril1 20666
      14.4  (Future - to be reviewed and classified)
            14.4.1  Planar incidence geometry   cplig 20672
            14.4.2  Algebra preliminaries   crpm 20677
            14.4.3  Transitive closure   ctcl 20679
PART 15  DEPRECATED SECTIONS
      15.1  Additional material on Group theory
            15.1.1  Definitions and basic properties for groups   cgr 20683
            15.1.2  Definition and basic properties of Abelian groups   cablo 20778
            15.1.3  Subgroups   csubgo 20798
            15.1.4  Operation properties   cass 20809
            15.1.5  Group-like structures   cmagm 20815
            15.1.6  Examples of Abelian groups   ablosn 20844
            15.1.7  Group homomorphism and isomorphism   cghom 20854
      15.2  Additional material on Rings and Fields
            15.2.1  Definition and basic properties   crngo 20872
            15.2.2  Examples of rings   cnrngo 20900
            15.2.3  Division Rings   cdrng 20902
            15.2.4  Star Fields   csfld 20905
            15.2.5  Fields and Rings   ccm2 20907
      15.3  Complex vector spaces
            15.3.1  Definition and basic properties   cvc 20931
            15.3.2  Examples of complex vector spaces   cncvc 20969
      15.4  Normed complex vector spaces
            15.4.1  Definition and basic properties   cnv 20970
            15.4.2  Examples of normed complex vector spaces   cnnv 21075
            15.4.3  Induced metric of a normed complex vector space   imsval 21084
            15.4.4  Inner product   cdip 21103
            15.4.5  Subspaces   css 21127
      15.5  Operators on complex vector spaces
            15.5.1  Definitions and basic properties   clno 21148
      15.6  Inner product (pre-Hilbert) spaces
            15.6.1  Definition and basic properties   ccphlo 21220
            15.6.2  Examples of pre-Hilbert spaces   cncph 21227
            15.6.3  Properties of pre-Hilbert spaces   isph 21230
      15.7  Complex Banach spaces
            15.7.1  Definition and basic properties   ccbn 21271
            15.7.2  Examples of complex Banach spaces   cnbn 21278
            15.7.3  Uniform Boundedness Theorem   ubthlem1 21279
            15.7.4  Minimizing Vector Theorem   minvecolem1 21283
      15.8  Complex Hilbert spaces
            15.8.1  Definition and basic properties   chlo 21294
            15.8.2  Standard axioms for a complex Hilbert space   hlex 21307
            15.8.3  Examples of complex Hilbert spaces   cnchl 21325
            15.8.4  Subspaces   ssphl 21326
            15.8.5  Hellinger-Toeplitz Theorem   htthlem 21327
      15.9  Hilbert Space Explorer
            15.9.1  Basic Hilbert space definitions   chil 21329
            15.9.2  Preliminary ZFC lemmas   df-hnorm 21378
            15.9.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 21391
            15.9.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 21409
            15.9.5  Vector operations   hvmulex 21421
            15.9.6  Inner product postulates for a Hilbert space   ax-hfi 21488
            15.9.7  Inner product   his5 21495
            15.9.8  Norms   dfhnorm2 21531
            15.9.9  Relate Hilbert space to normed complex vector spaces   hilablo 21569
            15.9.10  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 21588
            15.9.11  Cauchy sequences and limits   hcau 21593
            15.9.12  Derivation of the completeness axiom from ZF set theory   hilmet 21603
            15.9.13  Completeness postulate for a Hilbert space   ax-hcompl 21611
            15.9.14  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 21612
            15.9.15  Subspaces   df-sh 21616
            15.9.16  Closed subspaces   df-ch 21631
            15.9.17  Orthocomplements   df-oc 21661
            15.9.18  Subspace sum, span, lattice join, lattice supremum   df-shs 21717
            15.9.19  Projection theorem   pjhthlem1 21800
            15.9.20  Projectors   df-pjh 21804
            15.9.21  Orthomodular law   omlsilem 21811
            15.9.22  Projectors (cont.)   pjhtheu2 21825
            15.9.23  Hilbert lattice operations   sh0le 21849
            15.9.24  Span (cont.) and one-dimensional subspaces   spansn0 21950
            15.9.25  Operator sum, difference, and scalar multiplication   df-hosum 21992
            15.9.26  Commutes relation for Hilbert lattice elements   df-cm 22010
            15.9.27  Foulis-Holland theorem   fh1 22045
            15.9.28  Quantum Logic Explorer axioms   qlax1i 22054
            15.9.29  Orthogonal subspaces   chscllem1 22064
            15.9.30  Orthoarguesian laws 5OA and 3OA   5oalem1 22081
            15.9.31  Projectors (cont.)   pjorthi 22096
            15.9.32  Mayet's equation E_3   mayete3i 22155
            15.9.33  Zero and identity operators   df-h0op 22158
            15.9.34  Operations on Hilbert space operators   hoaddcl 22168
            15.9.35  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 22249
            15.9.36  Linear and continuous functionals and norms   df-nmfn 22255
            15.9.37  Adjoint   df-adjh 22259
            15.9.38  Dirac bra-ket notation   df-bra 22260
            15.9.39  Positive operators   df-leop 22262
            15.9.40  Eigenvectors, eigenvalues, spectrum   df-eigvec 22263
            15.9.41  Theorems about operators and functionals   nmopval 22266
            15.9.42  Riesz lemma   riesz3i 22472
            15.9.43  Adjoints (cont.)   cnlnadjlem1 22477
            15.9.44  Quantum computation error bound theorem   unierri 22514
            15.9.45  Dirac bra-ket notation (cont.)   branmfn 22515
            15.9.46  Positive operators (cont.)   leopg 22532
            15.9.47  Projectors as operators   pjhmopi 22556
            15.9.48  States on a Hilbert lattice   df-st 22621
            15.9.49  Godowski's equation   golem1 22681
            15.9.50  Covers relation; modular pairs   df-cv 22689
            15.9.51  Atoms   df-at 22748
            15.9.52  Superposition principle   superpos 22764
            15.9.53  Atoms, exchange and covering properties, atomicity   chcv1 22765
            15.9.54  Irreducibility   chirredlem1 22800
            15.9.55  Atoms (cont.)   atcvat3i 22806
            15.9.56  Modular symmetry   mdsymlem1 22813
PART 16  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      16.1  Mathboxes for user contributions
            16.1.1  Mathbox guidelines   mathbox 22852
      16.2  Mathbox for Stefan Allan
      16.3  Mathbox for Mario Carneiro
            16.3.1  Miscellaneous stuff   quartfull 22857
            16.3.2  Zeta function   czeta 22858
            16.3.3  Gamma function   clgam 22861
            16.3.4  Derangements and the Subfactorial   deranglem 22868
            16.3.5  The Erdős-Szekeres theorem   erdszelem1 22893
            16.3.6  The Kuratowski closure-complement theorem   kur14lem1 22908
            16.3.7  Retracts and sections   cretr 22919
            16.3.8  Path-connected and simply connected spaces   cpcon 22921
            16.3.9  Covering maps   ccvm 22957
            16.3.10  Undirected multigraphs   cumg 23031
            16.3.11  Normal numbers   snmlff 23083
            16.3.12  Godel-sets of formulas   cgoe 23087
            16.3.13  Models of ZF   cgze 23115
            16.3.14  Splitting fields   citr 23129
            16.3.15  p-adic number fields   czr 23145
      16.4  Mathbox for Paul Chapman
            16.4.1  Group homomorphism and isomorphism   ghomgrpilem1 23163
            16.4.2  Real and complex numbers (cont.)   climuzcnv 23175
            16.4.3  Miscellaneous theorems   elfzm12 23179
      16.5  Mathbox for Drahflow
      16.6  Mathbox for Scott Fenton
            16.6.1  ZFC Axioms in primitive form   axextprim 23218
            16.6.2  Untangled classes   untelirr 23225
            16.6.3  Extra propositional calculus theorems   3orel1 23232
            16.6.4  Misc. Useful Theorems   nepss 23243
            16.6.5  Properties of reals and complexes   sqdivzi 23249
            16.6.6  Greatest common divisor and divisibility   pdivsq 23272
            16.6.7  Properties of relationships   brtp 23276
            16.6.8  Properties of functions and mappings   funpsstri 23289
            16.6.9  Epsilon induction   setinds 23302
            16.6.10  Ordinal numbers   elpotr 23305
            16.6.11  Defined equality axioms   axextdfeq 23322
            16.6.12  Hypothesis builders   hbntg 23330
            16.6.13  The Predecessor Class   cpred 23335
            16.6.14  (Trans)finite Recursion Theorems   tfisg 23372
            16.6.15  Well-founded induction   tz6.26 23373
            16.6.16  Transitive closure under a relationship   ctrpred 23388
            16.6.17  Founded Induction   frmin 23410
            16.6.18  Ordering Ordinal Sequences   orderseqlem 23420
            16.6.19  Well-founded recursion   wfr3g 23423
            16.6.20  Transfinite recursion via Well-founded recursion   tfrALTlem 23444
            16.6.21  Founded Recursion   frr3g 23448
            16.6.22  Surreal Numbers   csur 23462
            16.6.23  Surreal Numbers: Ordering   axsltsolem1 23489
            16.6.24  Surreal Numbers: Birthday Function   axbday 23496
            16.6.25  Surreal Numbers: Density   axdenselem1 23503
            16.6.26  Surreal Numbers: Full-Eta Property   axfelem1 23514
            16.6.27  Symmetric difference   csymdif 23536
            16.6.28  Quantifier-free definitions   ctxp 23548
            16.6.29  Alternate ordered pairs   caltop 23664
            16.6.30  Tarskian geometry   cee 23690
            16.6.31  Tarski's axioms for geometry   axdimuniq 23715
            16.6.32  Congruence properties   cofs 23779
            16.6.33  Betweenness properties   btwntriv2 23809
            16.6.34  Segment Transportation   ctransport 23826
            16.6.35  Properties relating betweenness and congruence   cifs 23832
            16.6.36  Connectivity of betweenness   btwnconn1lem1 23884
            16.6.37  Segment less than or equal to   csegle 23903
            16.6.38  Outside of relationship   coutsideof 23916
            16.6.39  Lines and Rays   cline2 23931
            16.6.40  Bernoulli polynomials and sums of k-th powers   cbp 23955
            16.6.41  Rank theorems   rankung 23970
            16.6.42  Hereditarily Finite Sets   chf 23976
      16.7  Mathbox for Anthony Hart
            16.7.1  Propositional Calculus   tb-ax1 23991
            16.7.2  Predicate Calculus   quantriv 24013
            16.7.3  Misc. Single Axiom Systems   meran1 24024
            16.7.4  Connective Symmetry   negsym1 24030
      16.8  Mathbox for Chen-Pang He
            16.8.1  Ordinal topology   ontopbas 24041
      16.9  Mathbox for Jeff Hoffman
            16.9.1  Inferences for finite induction on generic function values   fveleq 24064
            16.9.2  gdc.mm   nnssi2 24068
      16.10  Mathbox for Wolf Lammen
      16.11  Mathbox for Frédéric Liné
            16.11.1  Theorems from other workspaces   tpssg 24097
            16.11.2  Propositional and predicate calculus   neleq12d 24098
            16.11.3  Linear temporal logic   wbox 24135
            16.11.4  Operations   ssoprab2g 24197
            16.11.5  General Set Theory   uninqs 24204
            16.11.6  The "maps to" notation   cmpfun 24308
            16.11.7  Cartesian Products   cpro 24316
            16.11.8  Operations on subsets and functions   ccst 24338
            16.11.9  Arithmetic   3timesi 24344
            16.11.10  Lattice (algebraic definition)   clatalg 24347
            16.11.11  Currying and Partial Mappings   ccur1 24360
            16.11.12  Order theory (Extensible Structure Builder)   corhom 24373
            16.11.13  Order theory   cpresetrel 24381
            16.11.14  Finite composites ( i. e. finite sums, products ... )   cprd 24464
            16.11.15  Operation properties   ccm1 24497
            16.11.16  Groups and related structures   ridlideq 24501
            16.11.17  Free structures   csubsmg 24549
            16.11.18  Translations   trdom2 24557
            16.11.19  Fields and Rings   com2i 24582
            16.11.20  Ideals   cidln 24609
            16.11.21  Generic modules and vector spaces (New Structure builder)   cact 24613
            16.11.22  Generic modules and vector spaces   cvec 24615
            16.11.23  Real vector spaces   cvr 24655
            16.11.24  Matrices   cmmat 24659
            16.11.25  Affine spaces   craffsp 24665
            16.11.26  Intervals of reals and extended reals   bsi 24667
            16.11.27  Topology   topnem 24678
            16.11.28  Continuous functions   cnrsfin 24691
            16.11.29  Homeomorphisms   dmhmph 24699
            16.11.30  Initial and final topologies   intopcoaconlem3b 24704
            16.11.31  Filters   efilcp 24718
            16.11.32  Limits   plimfil 24724
            16.11.33  Uniform spaces   cunifsp 24751
            16.11.34  Separated spaces: T0, T1, T2 (Hausdorff) ...   hst1 24753
            16.11.35  Compactness   indcomp 24755
            16.11.36  Connectedness   singempcon 24759
            16.11.37  Topological fields   ctopfld 24763
            16.11.38  Standard topology on RR   intrn 24765
            16.11.39  Standard topology of intervals of RR   stoi 24767
            16.11.40  Cantor's set   cntrset 24768
            16.11.41  Pre-calculus and Cartesian geometry   dmse1 24769
            16.11.42  Extended Real numbers   nolimf 24785
            16.11.43  ( RR ^ N ) and ( CC ^ N )   cplcv 24810
            16.11.44  Calculus   cintvl 24862
            16.11.45  Directed multi graphs   cmgra 24874
            16.11.46  Category and deductive system underlying "structure"   calg 24877
            16.11.47  Deductive systems   cded 24900
            16.11.48  Categories   ccatOLD 24918
            16.11.49  Homsets   chomOLD 24951
            16.11.50  Monomorphisms, Epimorphisms, Isomorphisms   cepiOLD 24969
            16.11.51  Functors   cfuncOLD 24997
            16.11.52  Subcategories   csubcat 25009
            16.11.53  Terminal and initial objects   ciobj 25026
            16.11.54  Sources and sinks   csrce 25031
            16.11.55  Limits and co-limits   clmct 25040
            16.11.56  Product and sum of two objects   cprodo 25043
            16.11.57  Tarski's classes   ctar 25047
            16.11.58  Category Set   ccmrcase 25076
            16.11.59  Grammars, Logics, Machines and Automata   ckln 25146
            16.11.60  Words   cwrd 25147
            16.11.61  Planar geometry   cpoints 25222
      16.12  Mathbox for Jeff Hankins
            16.12.1  Miscellany   a1i13 25366
            16.12.2  Basic topological facts   topbnd 25408
            16.12.3  Topology of the real numbers   reconnOLD 25421
            16.12.4  Refinements   cfne 25425
            16.12.5  Neighborhood bases determine topologies   neibastop1 25474
            16.12.6  Lattice structure of topologies   topmtcl 25478
            16.12.7  Filter bases   fgmin 25485
            16.12.8  Directed sets, nets   tailfval 25487
      16.13  Mathbox for Jeff Madsen
            16.13.1  Logic and set theory   anim12da 25498
            16.13.2  Real and complex numbers; integers   fimaxreOLD 25596
            16.13.3  Sequences and sums   sdclem2 25618
            16.13.4  Topology   unopnOLD 25630
            16.13.5  Metric spaces   metf1o 25635
            16.13.6  Continuous maps and homeomorphisms   constcncf 25644
            16.13.7  Product topologies   txtopiOLD 25652
            16.13.8  Boundedness   ctotbnd 25656
            16.13.9  Isometries   cismty 25688
            16.13.10  Heine-Borel Theorem   heibor1lem 25699
            16.13.11  Banach Fixed Point Theorem   bfplem1 25712
            16.13.12  Euclidean space   crrn 25715
            16.13.13  Intervals (continued)   ismrer1 25728
            16.13.14  Groups and related structures   exidcl 25732
            16.13.15  Rings   rngonegcl 25742
            16.13.16  Ring homomorphisms   crnghom 25757
            16.13.17  Commutative rings   ccring 25786
            16.13.18  Ideals   cidl 25798
            16.13.19  Prime rings and integral domains   cprrng 25837
            16.13.20  Ideal generators   cigen 25850
      16.14  Mathbox for Rodolfo Medina
            16.14.1  Partitions   prtlem60 25869
      16.15  Mathbox for Stefan O'Rear
            16.15.1  Additional elementary logic and set theory   nelss 25917
            16.15.2  Additional theory of functions   fninfp 25920
            16.15.3  Extensions beyond function theory   gsumvsmul 25930
            16.15.4  Additional topology   elrfi 25935
            16.15.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 25939
            16.15.6  Algebraic closure systems   cnacs 25943
            16.15.7  Miscellanea 1. Map utilities   constmap 25954
            16.15.8  Miscellanea for polynomials   ofmpteq 25963
            16.15.9  Multivariate polynomials over the integers   cmzpcl 25965
            16.15.10  Miscellanea for Diophantine sets 1   coeq0 25997
            16.15.11  Diophantine sets 1: definitions   cdioph 26000
            16.15.12  Diophantine sets 2 miscellanea   ellz1 26012
            16.15.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26018
            16.15.14  Diophantine sets 3: construction   diophrex 26021
            16.15.15  Diophantine sets 4 miscellanea   2sbcrex 26030
            16.15.16  Diophantine sets 4: Quantification   rexrabdioph 26041
            16.15.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26048
            16.15.18  Diophantine sets 6 miscellanea   fz1ssnn 26058
            16.15.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26060
            16.15.20  Pigeonhole Principle and cardinality helpers   fphpd 26065
            16.15.21  A non-closed set of reals is infinite   rencldnfilem 26069
            16.15.22  Miscellanea for Lagrange's theorem   icodiamlt 26071
            16.15.23  Lagrange's rational approximation theorem   irrapxlem1 26073
            16.15.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26080
            16.15.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26087
            16.15.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26129
            16.15.27  Logarithm laws generalized to an arbitrary base   reglogcl 26141
            16.15.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26149
            16.15.29  X and Y sequences 1: Definition and recurrence laws   crmx 26151
            16.15.30  Ordering and induction lemmas for the integers   monotuz 26192
            16.15.31  X and Y sequences 2: Order properties   rmxypos 26200
            16.15.32  Congruential equations   congtr 26218
            16.15.33  Alternating congruential equations   acongid 26228
            16.15.34  Additional theorems on integer divisibility   bezoutr 26238
            16.15.35  X and Y sequences 3: Divisibility properties   jm2.18 26247
            16.15.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26264
            16.15.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26274
            16.15.38  Uncategorized stuff not associated with a major project   setindtr 26283
            16.15.39  More equivalents of the Axiom of Choice   axac10 26292
            16.15.40  Finitely generated left modules   clfig 26331
            16.15.41  Noetherian left modules I   clnm 26339
            16.15.42  Addenda for structure powers   pwssplit0 26353
            16.15.43  Direct sum of left modules   cdsmm 26363
            16.15.44  Free modules   cfrlm 26378
            16.15.45  Every set admits a group structure iff choice   unxpwdom3 26422
            16.15.46  Independent sets and families   clindf 26440
            16.15.47  Characterization of free modules   lmimlbs 26472
            16.15.48  Noetherian rings and left modules II   clnr 26479
            16.15.49  Hilbert's Basis Theorem   cldgis 26491
            16.15.50  Additional material on polynomials [DEPRECATED]   cmnc 26501
            16.15.51  Degree and minimal polynomial of algebraic numbers   cdgraa 26511
            16.15.52  Algebraic integers I   citgo 26528
            16.15.53  Finite cardinality [SO]   en1uniel 26546
            16.15.54  Words in monoids and ordered group sum   issubmd 26549
            16.15.55  Transpositions in the symmetric group   cpmtr 26550
            16.15.56  The sign of a permutation   cpsgn 26580
            16.15.57  The matrix algebra   cmmul 26605
            16.15.58  The determinant   cmdat 26649
            16.15.59  Endomorphism algebra   cmend 26655
            16.15.60  Subfields   csdrg 26669
            16.15.61  Cyclic groups and order   idomrootle 26677
            16.15.62  Cyclotomic polynomials   ccytp 26687
            16.15.63  Miscellaneous topology   fgraphopab 26695
      16.16  Mathbox for Steve Rodriguez
            16.16.1  Miscellanea   iso0 26702
            16.16.2  Function operations   caofcan 26706
            16.16.3  Calculus   lhe4.4ex1a 26712
      16.17  Mathbox for Andrew Salmon
            16.17.1  Principia Mathematica * 10   pm10.12 26719
            16.17.2  Principia Mathematica * 11   2alanimi 26733
            16.17.3  Predicate Calculus   sbeqal1 26763
            16.17.4  Principia Mathematica * 13 and * 14   pm13.13a 26774
            16.17.5  Set Theory   elnev 26805
            16.17.6  Arithmetic   addcomgi 26828
            16.17.7  Geometry   cplusr 26829
      16.18  Mathbox for Jarvin Udandy
      16.19  Mathbox for David A. Wheeler
            16.19.1  Natural deduction   19.8ad 26876
            16.19.2  Greater than, greater than or equal to.   cge-real 26879
            16.19.3  Hyperbolic trig functions   csinh 26889
            16.19.4  Reciprocal trig functions (sec, csc, cot)   csec 26900
            16.19.5  Identities for "if"   ifnmfalse 26922
            16.19.6  Not-member-of   AnelBC 26923
            16.19.7  Decimal point   cdp2 26924
            16.19.8  Signum (sgn or sign) function   csgn 26932
            16.19.9  Ceiling function   ccei 26942
            16.19.10  Logarithm laws generalized to an arbitrary base   clogb 26946
            16.19.11  Miscellaneous   2m1e1 26951
      16.20  Mathbox for Alan Sare
            16.20.1  Conventional Metamath proofs, some derived from VD proofs   iidn3 26955
            16.20.2  What is Virtual Deduction?   wvd1 27030
            16.20.3  Virtual Deduction Theorems   df-vd1 27031
            16.20.4  Theorems proved using virtual deduction   trsspwALT 27282
            16.20.5  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 27312
            16.20.6  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 27379
            16.20.7  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 27383
            16.20.8  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 27390
            16.20.9  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 27393
      16.21  Mathbox for Jonathan Ben-Naim
            16.21.1  First order logic and set theory   bnj170 27412
            16.21.2  Well founded induction and recursion   bnj110 27579
            16.21.3  The existence of a minimal element in certain classes   bnj69 27729
            16.21.4  Well-founded induction   bnj1204 27731
            16.21.5  Well-founded recursion, part 1 of 3   bnj60 27781
            16.21.6  Well-founded recursion, part 2 of 3   bnj1500 27787
            16.21.7  Well-founded recursion, part 3 of 3   bnj1522 27791
      16.22  Mathbox for Norm Megill
            16.22.1  Obsolete experiments to study ax-12o   ax12-2 27792
            16.22.2  Miscellanea   cnaddcom 27850
            16.22.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 27853
            16.22.4  Functionals and kernels of a left vector space (or module)   clfn 27936
            16.22.5  Opposite rings and dual vector spaces   cld 28002
            16.22.6  Ortholattices and orthomodular lattices   cops 28051
            16.22.7  Atomic lattices with covering property   ccvr 28141
            16.22.8  Hilbert lattices   chlt 28229
            16.22.9  Projective geometries based on Hilbert lattices   clln 28369
            16.22.10  Construction of a vector space from a Hilbert lattice   cdlema1N 28669
            16.22.11  Construction of involution and inner product from a Hilbert lattice   clpoN 30359

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