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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  Humor
      14.3  (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 4
            1.3.2  The axioms of propositional calculus   ax-1 6
            1.3.3  Logical implication   mp2b 10
            1.3.4  Logical negation   con4d 98
            1.3.5  Logical equivalence   wb 175
            1.3.6  Logical disjunction and conjunction   wo 356
            1.3.7  Miscellaneous theorems of propositional calculus   pm5.21nd 867
            1.3.8  Abbreviated conjunction and disjunction of three wff's   w3o 932
            1.3.9  Logical 'nand' (Sheffer stroke)   wnan 1286
            1.3.10  Logical 'xor'   wxo 1294
            1.3.11  True and false constants   wtru 1306
            1.3.12  Truth tables   truantru 1321
            1.3.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1347
            1.3.14  Half-adders and full adders in propositional calculus   whad 1363
      1.4  Other axiomatizations of classical propositional calculus
            1.4.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1389
            1.4.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1407
            1.4.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1418
            1.4.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1424
            1.4.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1443
            1.4.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1447
            1.4.7  Deriving the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1462
            1.4.8  Deriving the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1485
            1.4.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1498
            1.4.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1517
      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 1521
            1.5.2  Introduce equality axioms ax-8, ax-11, ax-13, and ax-14   cv 1607
            1.5.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1617
            1.5.4  Introduce equality axioms ax-9v and ax-12   ax-9v 1621
            1.5.5  Derive ax-12o from ax-12   ax12o10lem1 1624
            1.5.6  Derive ax-10   ax10lem16 1654
            1.5.7  Derive ax-9 from the weaker version ax-9v   ax9 1672
            1.5.8  Introduce Axiom of Existence ax-9   ax-9 1673
            1.5.9  Derive ax-4, ax-5o, and ax-6o   ax4 1680
            1.5.10  "Pure" predicate calculus including ax-4, without distinct variables   a4i 1688
            1.5.11  Equality theorems without distinct variables   ax9o 1803
            1.5.12  Axioms ax-10 and ax-11   ax10o 1823
            1.5.13  Substitution (without distinct variables)   wsb 1871
            1.5.14  Theorems using axiom ax-11   equs5a 1900
      1.6  Predicate calculus with distinct variables
            1.6.1  Derive the axiom of distinct variables ax-16   a4imv 1911
            1.6.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1927
            1.6.3  Theorems without distinct variables that use axiom ax-11o   ax11b 1930
            1.6.4  Predicate calculus with distinct variables (cont.)   ax11v 1978
            1.6.5  More substitution theorems   equsb3lem 2049
            1.6.6  Existential uniqueness   weu 2102
      1.7  Other axiomatizations related to classical predicate calculus
            1.7.1  Predicate calculus with all distinct variables   ax-7d 2192
            1.7.2  Aristotelian logic: Assertic syllogisms   barbara 2198
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 2222
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2227
            2.1.3  Class form not-free predicate   wnfc 2360
            2.1.4  Negated equality and membership   wne 2400
            2.1.5  Restricted quantification   wral 2495
            2.1.6  The universal class   cvv 2712
            2.1.7  Conditional equality (experimental)   wcdeq 2889
            2.1.8  Russell's Paradox   ru 2905
            2.1.9  Proper substitution of classes for sets   wsbc 2906
            2.1.10  Proper substitution of classes for sets into classes   csb 2989
            2.1.11  Define basic set operations and relations   cdif 3055
            2.1.12  Subclasses and subsets   df-ss 3069
            2.1.13  The difference, union, and intersection of two classes   difeq1 3184
            2.1.14  The empty set   c0 3342
            2.1.15  "Weak deduction theorem" for set theory   cif 3450
            2.1.16  Power classes   cpw 3510
            2.1.17  Unordered and ordered pairs   csn 3524
            2.1.18  The union of a class   cuni 3707
            2.1.19  The intersection of a class   cint 3740
            2.1.20  Indexed union and intersection   ciun 3783
            2.1.21  Disjointness   wdisj 3871
            2.1.22  Binary relations   wbr 3900
            2.1.23  Ordered-pair class abstractions (class builders)   copab 3953
            2.1.24  Transitive classes   wtr 3989
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4007
            2.2.2  Derive the Axiom of Separation   axsep 4016
            2.2.3  Derive the Null Set Axiom   zfnuleu 4022
            2.2.4  Theorems requiring subset and intersection existence   nalset 4027
            2.2.5  Theorems requiring empty set existence   class2set 4051
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4061
            2.3.2  Derive the Axiom of Pairing   zfpair 4085
            2.3.3  Ordered pair theorem   opnz 4114
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4143
            2.3.5  Power class of union and intersection   pwin 4169
            2.3.6  Epsilon and identity relations   cep 4175
            2.3.7  Partial and complete ordering   wpo 4184
            2.3.8  Founded and well-ordering relations   wfr 4221
            2.3.9  Ordinals   word 4263
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4382
            2.4.2  Ordinals (continued)   ordon 4444
            2.4.3  Transfinite induction   tfi 4514
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4526
            2.4.5  Peano's postulates   peano1 4545
            2.4.6  Finite induction (for finite ordinals)   find 4551
            2.4.7  Functions and relations   cxp 4557
            2.4.8  Operations   co 5685
            2.4.9  "Maps to" notation   elmpt2cl 5887
            2.4.10  Function operation   cof 5902
            2.4.11  First and second members of an ordered pair   c1st 5946
            2.4.12  Function transposition   ctpos 6059
            2.4.13  Curry and uncurry   ccur 6098
            2.4.14  Proper subset relation   crpss 6102
            2.4.15  Definite description binder (inverted iota)   cio 6115
            2.4.16  Cantor's Theorem   canth 6152
            2.4.17  Undefined values and restricted iota (description binder)   cund 6154
            2.4.18  Functions on ordinals; strictly monotone ordinal functions   iunon 6215
            2.4.19  "Strong" transfinite recursion   crecs 6247
            2.4.20  Recursive definition generator   crdg 6282
            2.4.21  Finite recursion   frfnom 6307
            2.4.22  Abian's "most fundamental" fixed point theorem   abianfplem 6330
            2.4.23  Ordinal arithmetic   c1o 6332
            2.4.24  Natural number arithmetic   nna0 6462
            2.4.25  Equivalence relations and classes   wer 6517
            2.4.26  The mapping operation   cmap 6632
            2.4.27  Infinite Cartesian products   cixp 6677
            2.4.28  Equinumerosity   cen 6720
            2.4.29  Schroeder-Bernstein Theorem   sbthlem1 6830
            2.4.30  Equinumerosity (cont.)   xpf1o 6882
            2.4.31  Pigeonhole Principle   phplem1 6899
            2.4.32  Finite sets   onomeneq 6909
            2.4.33  Finite intersections   cfi 7022
            2.4.34  Hall's marriage theorem   marypha1lem 7044
            2.4.35  Supremum   csup 7051
            2.4.36  Ordinal isomorphism, Hartog's theorem   coi 7082
            2.4.37  Hartogs function, order types, weak dominance   char 7128
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7164
            2.5.2  Axiom of Infinity equivalents   inf0 7180
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7197
            2.6.2  Existence of omega (the set of natural numbers)   omex 7202
            2.6.3  Cantor normal form   ccnf 7220
            2.6.4  Transitive closure   trcl 7268
            2.6.5  Rank   cr1 7292
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7413
            2.6.7  Cardinal numbers   ccrd 7426
            2.6.8  Axiom of Choice equivalents   wac 7600
            2.6.9  Cardinal number arithmetic   ccda 7651
            2.6.10  The Ackermann bijection   ackbij2lem1 7703
            2.6.11  Cofinality (without Axiom of Choice)   cflem 7730
            2.6.12  Eight inequivalent definitions of finite set   sornom 7761
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 7900
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 7943
            3.2.2  AC equivalents: well ordering, Zorn's lemma   numthcor 7979
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8026
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8050
            3.2.5  Cofinality using Axiom of Choice   alephreg 8058
      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 8158
            4.1.2  Weak universes   cwun 8176
            4.1.3  Tarski's classes   ctsk 8224
            4.1.4  Grothendieck's universes   cgru 8266
      4.2  ZFC Set Theory plus the Tarksi-Grothendieck Axiom
            4.2.1  Introduce the Tarksi-Grothendieck Axiom   ax-groth 8299
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8302
            4.2.3  Tarski map function   ctskm 8313
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 8320
            5.1.2  Final derivation of real and complex number postulates   axaddf 8621
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8647
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 8672
            5.2.2  Infinity and the extended real number system   cpnf 8718
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 8747
            5.2.4  Ordering on reals   lttr 8752
            5.2.5  Initial properties of the complex numbers   mul12 8831
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 8874
            5.3.2  Subtraction   cmin 8886
            5.3.3  Multiplication   muladd 9050
            5.3.4  Ordering on reals (cont.)   gt0ne0 9076
            5.3.5  Reciprocals   ixi 9233
            5.3.6  Division   cdiv 9255
            5.3.7  Ordering on reals (cont.)   elimgt0 9412
            5.3.8  Completeness Axiom and Suprema   fimaxre 9521
            5.3.9  Imaginary and complex number properties   inelr 9556
            5.3.10  Function operation analogue theorems   ofsubeq0 9563
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9566
            5.4.2  Principle of mathematical induction   nnind 9584
            5.4.3  Decimal representation of numbers   c2 9615
            5.4.4  Some properties of specific numbers   0p1e1 9659
            5.4.5  The Archimedean property   nnunb 9779
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 9783
            5.4.7  Integers (as a subset of complex numbers)   cz 9842
            5.4.8  Decimal arithmetic   cdc 9942
            5.4.9  Upper partititions of integers   cuz 10048
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10129
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10134
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10160
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10172
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10267
            5.5.3  Supremum on the extended reals   xrsupexmnf 10440
            5.5.4  Real number intervals   cioo 10473
            5.5.5  Finite intervals of integers   cfz 10597
            5.5.6  Half-open integer ranges   cfzo 10685
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 10739
            5.6.2  The modulo (remainder) operation   cmo 10788
            5.6.3  The infinite sequence builder "seq"   om2uz0i 10825
            5.6.4  Integer powers   cexp 10919
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11096
            5.6.6  Factorial function   cfa 11102
            5.6.7  The binomial coefficient operation   cbc 11129
            5.6.8  The ` # ` (finite set size) function   chash 11151
            5.6.9  Words over a set   cword 11216
            5.6.10  Longer string literals   cs2 11304
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11374
            5.7.2  Real and imaginary parts; conjugate   ccj 11394
            5.7.3  Square root; absolute value   csqr 11531
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 11755
            5.8.2  Limits   cli 11769
            5.8.3  Finite and infinite sums   csu 11969
            5.8.4  The binomial theorem   binomlem 12098
            5.8.5  Infinite sums (cont.)   isumshft 12106
            5.8.6  Miscellaneous converging and diverging sequences   divrcnv 12119
            5.8.7  Arithmetic series   arisum 12126
            5.8.8  Geometric series   expcnv 12130
            5.8.9  Ratio test for infinite series convergence   cvgrat 12147
            5.8.10  Mertens' theorem   mertenslem1 12148
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12151
            5.9.2  _e is irrational   eirrlem 12290
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12295
            5.10.2  The reals are uncountable   rpnnen2lem1 12301
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12334
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12337
            6.1.3  The divides relation   cdivides 12339
            6.1.4  The division algorithm   divalglem0 12400
            6.1.5  Bit sequences   cbits 12418
            6.1.6  The greatest common divisor operator   cgcd 12493
            6.1.7  Bézout's identity   bezoutlem1 12525
            6.1.8  Algorithms   nn0seqcvgd 12548
            6.1.9  Euclid's Algorithm   eucalgval2 12559
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 12566
            6.2.2  Properties of the canonical representation of a rational   cnumer 12612
            6.2.3  Euler's theorem   codz 12639
            6.2.4  Pythagorean Triples   coprimeprodsq 12670
            6.2.5  The prime count function   cpc 12697
            6.2.6  Pocklington's theorem   prmpwdvds 12759
            6.2.7  Infinite primes theorem   unbenlem 12763
            6.2.8  Sum of prime reciprocals   prmreclem1 12771
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 12778
            6.2.10  Lagrange's four-square theorem   cgz 12784
            6.2.11  Van der Waerden's theorem   cvdwa 12820
            6.2.12  Ramsey's theorem   cram 12854
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 12886
            6.2.14  Specific prime numbers   4nprm 12914
            6.2.15  Very large primes   1259lem1 12937
PART 7  EXTENSIBLE STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 12952
            7.1.2  Slot definitions   cplusg 13016
            7.1.3  Definition of the structure product   crest 13133
            7.1.4  Definition of the structure quotient   cordt 13206
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13314
            7.2.2  Algebraic closure systems   isacs 13332
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13344
            8.1.2  Opposite category   coppc 13390
            8.1.3  Monomorphisms and epimorphisms   cmon 13404
            8.1.4  Sections, inverses, isomorphisms   csect 13420
            8.1.5  Subcategories   cssc 13456
            8.1.6  Functors   cfunc 13500
            8.1.7  Natural transformations and the functor category   cnat 13542
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 13618
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 13640
            8.3.2  The category of categories   ccatc 13658
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 13672
            8.4.2  Functor evaluation   cevlf 13712
            8.4.3  Hom functor   chof 13740
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 13758
            9.2.2  Lattices   clat 13835
            9.2.3  The dual of an ordered set   codu 13916
            9.2.4  Subset order structures   cipo 13938
            9.2.5  Distributive lattices   latmass 13966
            9.2.6  Posets and lattices as relations   cps 13976
            9.2.7  Directed sets, nets   cdir 14025
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14036
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14088
            10.1.3  Ordered group sum operation   gsumvallem1 14123
            10.1.4  Free monoids   cfrmd 14144
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14164
            10.2.2  Subgroups and Quotient groups   csubg 14290
            10.2.3  Elementary theory of group homomorphisms   cghm 14355
            10.2.4  Isomorphisms of groups   cgim 14396
            10.2.5  Group actions   cga 14418
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 14444
            10.2.7  Centralizers and centers   ccntz 14466
            10.2.8  The opposite group   coppg 14493
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 14515
            10.2.10  Direct products   clsm 14620
            10.2.11  Free groups   cefg 14690
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 14764
            10.3.2  Cyclic groups   ccyg 14839
            10.3.3  Group sum operation   gsumval3a 14864
            10.3.4  Internal direct products   cdprd 14906
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 14975
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15000
            10.4.2  Definition and basic properties   crg 15012
            10.4.3  Opposite ring   coppr 15079
            10.4.4  Divisibility   cdsr 15095
            10.4.5  Ring homomorphisms   crh 15169
      10.5  Division rings and Fields
            10.5.1  Definition and basic properties   cdr 15187
            10.5.2  Subrings of a ring   csubrg 15216
            10.5.3  Absolute value (abstract algebra)   cabv 15256
            10.5.4  Star rings   cstf 15283
      10.6  Left Modules
            10.6.1  Definition and basic properties   clmod 15302
            10.6.2  Subspaces and spans in a left module   clss 15364
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 15451
            10.6.4  Subspace sum; bases for a left module   clbs 15502
      10.7  Vector Spaces
            10.7.1  Definition and basic properties   clvec 15530
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 15593
            10.8.2  Two-sided ideals and quotient rings   c2idl 15655
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 15665
            10.8.4  Nonzero rings   cnzr 15681
            10.8.5  Left regular elements. More kinds of ring   crlreg 15692
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 15722
      10.10  Abstract Multivariate Polynomials
            10.10.1  Definition and basic properties   cmps 15759
            10.10.2  Polynomial evaluation   evlslem4 15917
            10.10.3  Univariate Polynomials   cps1 15922
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16041
            10.11.2  Algebraic constructions based on the complexes   czrh 16123
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16200
            10.12.2  Orthocomplements and closed subspaces   cocv 16232
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16272
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16303
            11.1.2  TopBases for topologies   isbasisg 16357
            11.1.3  Examples of topologies   distop 16405
            11.1.4  Closure and interior   ccld 16425
            11.1.5  Neighborhoods   cnei 16506
            11.1.6  Limit points and perfect sets   clp 16538
            11.1.7  Subspace topologies   restrcl 16560
            11.1.8  Order topology   ordtbaslem 16590
            11.1.9  Limits and Continuity in topological spaces   ccn 16626
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 16706
            11.1.11  Compactness   ccmp 16785
            11.1.12  Connectedness   ccon 16809
            11.1.13  First- and second-countability   c1stc 16835
            11.1.14  Local topological properties   clly 16862
            11.1.15  Compactly generated spaces   ckgen 16900
            11.1.16  Product topologies   ctx 16927
            11.1.17  Continuous function-builders   cnmptid 17027
            11.1.18  Quotient maps and quotient topology   ckq 17056
            11.1.19  Homeomorphisms   chmeo 17116
      11.2  Filters and filter bases
            11.2.1  Filter Bases   cfbas 17190
            11.2.2  Filters   cfil 17212
            11.2.3  Ultrafilters   cufil 17266
            11.2.4  Filter limits   cfm 17300
            11.2.5  Topological groups   ctmd 17425
            11.2.6  Infinite group sum on topological groups   ctsu 17480
            11.2.7  Topological rings, fields, vector spaces   ctrg 17510
      11.3  Metric spaces
            11.3.1  Basic metric space properties   cxme 17554
            11.3.2  Metric space balls   blfval 17619
            11.3.3  Open sets of a metric space   mopnval 17656
            11.3.4  Continuity in metric spaces   metcnp3 17758
            11.3.5  Examples of metric spaces   dscmet 17767
            11.3.6  Normed algebraic structures   cnm 17771
            11.3.7  Normed space homomorphisms (bounded linear operators)   cnmo 17886
            11.3.8  Topology on the Reals   qtopbaslem 17939
            11.3.9  Topological definitions using the reals   cii 18051
            11.3.10  Path homotopy   chtpy 18137
            11.3.11  The fundamental group   cpco 18170
            11.3.12  Complex left modules   cclm 18232
            11.3.13  Complex pre-Hilbert space   ccph 18274
            11.3.14  Convergence and completeness   ccfil 18350
            11.3.15  Baire's Category Theorem   bcthlem1 18418
            11.3.16  Banach spaces and complex Hilbert spaces   ccms 18426
            11.3.17  Minimizing Vector Theorem   minveclem1 18460
            11.3.18  Projection theorem   pjthlem1 18473
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 18480
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 18494
            12.2.2  Lebesgue integration   cmbf 18641
      12.3  Derivatives
            12.3.1  Real and Complex Differentiation   climc 18884
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19069
            13.1.2  Polynomial degrees   cmdg 19111
            13.1.3  The division algorithm for univariate polynomials   cmn1 19183
            13.1.4  Elementary properties of complex polynomials   cply 19238
            13.1.5  The Division algorithm for polynomials   cquot 19342
            13.1.6  Algebraic numbers   caa 19366
            13.1.7  Liouville's approximation theorem   aalioulem1 19384
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 19404
            13.2.2  Uniform convergence   culm 19427
            13.2.3  Power series   pserval 19458
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 19491
            13.3.2  Properties of pi = 3.14159...   pilem1 19499
            13.3.3  Mapping of the exponential function   efgh 19570
            13.3.4  The natural logarithm on complex numbers   clog 19579
            13.3.5  Solutions of quardatic, cubic, and quartic equations   quad2 19779
            13.3.6  Inverse trigonometric functions   casin 19802
            13.3.7  The Birthday Problem   log2ublem1 19886
            13.3.8  Areas in R^2   carea 19894
            13.3.9  More miscellaneous converging sequences   rlimcnp 19904
            13.3.10  Inequality of arithmetic and geometric means   cvxcl 19923
            13.3.11  Euler-Mascheroni constant   cem 19930
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 19950
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 19954
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 19962
            13.4.4  Number-theoretical functions   ccht 19972
            13.4.5  Perfect Number Theorem   mersenne 20110
            13.4.6  Characters of Z/nZ   cdchr 20115
            13.4.7  Bertrand's postulate   bcctr 20158
            13.4.8  Legendre symbol   clgs 20177
            13.4.9  Quadratic Reciprocity   lgseisenlem1 20232
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 20246
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 20262
            13.4.12  The Prime Number Theorem   mudivsum 20323
            13.4.13  Ostrowski's theorem   abvcxp 20408
PART 14  MISCELLANEA
      14.1  Definitional Examples
      14.2  Humor
            14.2.1  April Fool's theorem   avril1 20461
      14.3  (Future - to be reviewed and classified)
            14.3.1  Planar incidence geometry   cplig 20467
            14.3.2  Algebra preliminaries   crpm 20472
            14.3.3  Transitive closure   ctcl 20474
PART 15  DEPRECATED SECTIONS
      15.1  Additional material on Group theory
            15.1.1  Definitions and basic properties for groups   cgr 20478
            15.1.2  Definition and basic properties of Abelian groups   cablo 20573
            15.1.3  Subgroups   csubgo 20593
            15.1.4  Operation properties   cass 20604
            15.1.5  Group-like structures   cmagm 20610
            15.1.6  Examples of Abelian groups   ablosn 20639
            15.1.7  Group homomorphism and isomorphism   cghom 20649
      15.2  Additional material on Rings and Fields
            15.2.1  Definition and basic properties   crngo 20667
            15.2.2  Examples of rings   cnrngo 20695
            15.2.3  Division Rings   cdrng 20697
            15.2.4  Star Fields   csfld 20700
            15.2.5  Fields and Rings   ccm2 20702
      15.3  Complex vector spaces
            15.3.1  Definition and basic properties   cvc 20726
            15.3.2  Examples of complex vector spaces   cncvc 20764
      15.4  Normed complex vector spaces
            15.4.1  Definition and basic properties   cnv 20765
            15.4.2  Examples of normed complex vector spaces   cnnv 20870
            15.4.3  Induced metric of a normed complex vector space   imsval 20879
            15.4.4  Inner product   cdip 20898
            15.4.5  Subspaces   css 20922
      15.5  Operators on complex vector spaces
            15.5.1  Definitions and basic properties   clno 20943
      15.6  Inner product (pre-Hilbert) spaces
            15.6.1  Definition and basic properties   ccphlo 21015
            15.6.2  Examples of pre-Hilbert spaces   cncph 21022
            15.6.3  Properties of pre-Hilbert spaces   isph 21025
      15.7  Complex Banach spaces
            15.7.1  Definition and basic properties   ccbn 21066
            15.7.2  Examples of complex Banach spaces   cnbn 21073
            15.7.3  Uniform Boundedness Theorem   ubthlem1 21074
            15.7.4  Minimizing Vector Theorem   minvecolem1 21078
      15.8  Complex Hilbert spaces
            15.8.1  Definition and basic properties   chlo 21089
            15.8.2  Standard axioms for a complex Hilbert space   hlex 21102
            15.8.3  Examples of complex Hilbert spaces   cnchl 21120
            15.8.4  Subspaces   ssphl 21121
            15.8.5  Hellinger-Toeplitz Theorem   htthlem 21122
      15.9  Hilbert Space Explorer
            15.9.1  Basic Hilbert space definitions   chil 21124
            15.9.2  Preliminary ZFC lemmas   df-hnorm 21173
            15.9.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 21186
            15.9.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 21204
            15.9.5  Vector operations   hvmulex 21216
            15.9.6  Inner product postulates for a Hilbert space   ax-hfi 21283
            15.9.7  Inner product   his5 21290
            15.9.8  Norms   dfhnorm2 21326
            15.9.9  Relate Hilbert space to normed complex vector spaces   hilablo 21364
            15.9.10  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 21383
            15.9.11  Cauchy sequences and limits   hcau 21388
            15.9.12  Derivation of the completeness axiom from ZF set theory   hilmet 21398
            15.9.13  Completeness postulate for a Hilbert space   ax-hcompl 21406
            15.9.14  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 21407
            15.9.15  Subspaces   df-sh 21411
            15.9.16  Closed subspaces   df-ch 21426
            15.9.17  Orthocomplements   df-oc 21456
            15.9.18  Subspace sum, span, lattice join, lattice supremum   df-shs 21512
            15.9.19  Projection theorem   pjhthlem1 21595
            15.9.20  Projectors   df-pjh 21599
            15.9.21  Orthomodular law   omlsilem 21606
            15.9.22  Projectors (cont.)   pjhtheu2 21620
            15.9.23  Hilbert lattice operations   sh0le 21644
            15.9.24  Span (cont.) and one-dimensional subspaces   spansn0 21745
            15.9.25  Operator sum, difference, and scalar multiplication   df-hosum 21787
            15.9.26  Commutes relation for Hilbert lattice elements   df-cm 21805
            15.9.27  Foulis-Holland theorem   fh1 21840
            15.9.28  Quantum Logic Explorer axioms   qlax1i 21849
            15.9.29  Orthogonal subspaces   chscllem1 21859
            15.9.30  Orthoarguesian laws 5OA and 3OA   5oalem1 21876
            15.9.31  Projectors (cont.)   pjorthi 21891
            15.9.32  Mayet's equation E_3   mayete3i 21950
            15.9.33  Zero and identity operators   df-h0op 21953
            15.9.34  Operations on Hilbert space operators   hoaddcl 21963
            15.9.35  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 22044
            15.9.36  Linear and continuous functionals and norms   df-nmfn 22050
            15.9.37  Adjoint   df-adjh 22054
            15.9.38  Dirac bra-ket notation   df-bra 22055
            15.9.39  Positive operators   df-leop 22057
            15.9.40  Eigenvectors, eigenvalues, spectrum   df-eigvec 22058
            15.9.41  Theorems about operators and functionals   nmopval 22061
            15.9.42  Riesz lemma   riesz3i 22267
            15.9.43  Adjoints (cont.)   cnlnadjlem1 22272
            15.9.44  Quantum computation error bound theorem   unierri 22309
            15.9.45  Dirac bra-ket notation (cont.)   branmfn 22310
            15.9.46  Positive operators (cont.)   leopg 22327
            15.9.47  Projectors as operators   pjhmopi 22351
            15.9.48  States on a Hilbert lattice   df-st 22416
            15.9.49  Godowski's equation   golem1 22476
            15.9.50  Covers relation; modular pairs   df-cv 22484
            15.9.51  Atoms   df-at 22543
            15.9.52  Superposition principle   superpos 22559
            15.9.53  Atoms, exchange and covering properties, atomicity   chcv1 22560
            15.9.54  Irreducibility   chirredlem1 22595
            15.9.55  Atoms (cont.)   atcvat3i 22601
            15.9.56  Modular symmetry   mdsymlem1 22608
PART 16  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      16.1  Mathboxes for user contributions
            16.1.1  Mathbox guidelines   mathbox 22647
      16.2  Mathbox for Stefan Allan
      16.3  Mathbox for Mario Carneiro
            16.3.1  Miscellaneous stuff   quartfull 22652
            16.3.2  Zeta function   czeta 22653
            16.3.3  Gamma function   clgam 22656
            16.3.4  Derangements and the Subfactorial   deranglem 22663
            16.3.5  The Erdős-Szekeres theorem   erdszelem1 22688
            16.3.6  The Kuratowski closure-complement theorem   kur14lem1 22703
            16.3.7  Retracts and sections   cretr 22714
            16.3.8  Path-connected and simply connected spaces   cpcon 22716
            16.3.9  Covering maps   ccvm 22752
            16.3.10  Undirected multigraphs   cumg 22826
            16.3.11  Normal numbers   snmlff 22878
            16.3.12  Godel-sets of formulas   cgoe 22882
            16.3.13  Models of ZF   cgze 22910
            16.3.14  Splitting fields   citr 22924
            16.3.15  p-adic number fields   czr 22940
      16.4  Mathbox for Paul Chapman
            16.4.1  Group homomorphism and isomorphism   ghomgrpilem1 22958
            16.4.2  Real and complex numbers (cont.)   climuzcnv 22970
            16.4.3  Miscellaneous theorems   elfzm12 22974
      16.5  Mathbox for Drahflow
      16.6  Mathbox for Scott Fenton
            16.6.1  ZFC Axioms in primitive form   axextprim 23013
            16.6.2  Untangled classes   untelirr 23020
            16.6.3  Extra propositional calculus theorems   3orel1 23027
            16.6.4  Misc. Useful Theorems   nepss 23038
            16.6.5  Properties of reals and complexes   sqdivzi 23044
            16.6.6  Greatest common divisor and divisibility   pdivsq 23072
            16.6.7  Properties of relationships   brtp 23076
            16.6.8  Properties of functions and mappings   funpsstri 23089
            16.6.9  Epsilon induction   setinds 23102
            16.6.10  Ordinal numbers   elpotr 23105
            16.6.11  Defined equality axioms   axextdfeq 23122
            16.6.12  Hypothesis builders   hbntg 23130
            16.6.13  The Predecessor Class   cpred 23135
            16.6.14  (Trans)finite Recursion Theorems   tfisg 23172
            16.6.15  Well-founded induction   tz6.26 23173
            16.6.16  Transitive closure under a relationship   ctrpred 23188
            16.6.17  Founded Induction   frmin 23210
            16.6.18  Ordering Ordinal Sequences   orderseqlem 23220
            16.6.19  Well-founded recursion   wfr3g 23223
            16.6.20  Transfinite recursion via Well-founded recursion   tfrALTlem 23244
            16.6.21  Founded Recursion   frr3g 23248
            16.6.22  Surreal Numbers   csur 23262
            16.6.23  Surreal Numbers: Ordering   axsltsolem1 23289
            16.6.24  Surreal Numbers: Birthday Function   axbday 23296
            16.6.25  Surreal Numbers: Density   axdenselem1 23303
            16.6.26  Surreal Numbers: Full-Eta Property   axfelem1 23314
            16.6.27  Symmetric difference   csymdif 23336
            16.6.28  Quantifier-free definitions   ctxp 23348
            16.6.29  Alternate ordered pairs   caltop 23464
            16.6.30  Tarskian geometry   cee 23490
            16.6.31  Tarski's axioms for geometry   axdimuniq 23515
            16.6.32  Congruence properties   cofs 23579
            16.6.33  Betweenness properties   btwntriv2 23609
            16.6.34  Segment Transportation   ctransport 23626
            16.6.35  Properties relating betweenness and congruence   cifs 23632
            16.6.36  Connectivity of betweenness   btwnconn1lem1 23684
            16.6.37  Segment less than or equal to   csegle 23703
            16.6.38  Outside of relationship   coutsideof 23716
            16.6.39  Lines and Rays   cline2 23731
            16.6.40  Bernoulli polynomials and sums of k-th powers   cbp 23755
            16.6.41  Rank theorems   rankung 23770
            16.6.42  Hereditarily Finite Sets   chf 23776
      16.7  Mathbox for Anthony Hart
            16.7.1  Propositional Calculus   tb-ax1 23791
            16.7.2  Predicate Calculus   quantriv 23813
            16.7.3  Misc. Single Axiom Systems   meran1 23824
            16.7.4  Connective Symmetry   negsym1 23830
      16.8  Mathbox for Chen-Pang He
            16.8.1  Ordinal topology   ontopbas 23841
      16.9  Mathbox for Jeff Hoffman
            16.9.1  Inferences for finite induction on generic function values   fveleq 23864
            16.9.2  gdc.mm   nnssi2 23868
      16.10  Mathbox for Wolf Lammen
      16.11  Mathbox for Frédéric Liné
            16.11.1  Theorems from other workspaces   tpssg 23897
            16.11.2  Propositional and predicate calculus   neleq12d 23898
            16.11.3  Linear temporal logic   wbox 23935
            16.11.4  Operations   ssoprab2g 23997
            16.11.5  General Set Theory   uninqs 24004
            16.11.6  The "maps to" notation   cmpfun 24108
            16.11.7  Cartesian Products   cpro 24116
            16.11.8  Operations on subsets and functions   ccst 24138
            16.11.9  Arithmetic   3timesi 24144
            16.11.10  Lattice (algebraic definition)   clatalg 24147
            16.11.11  Currying and Partial Mappings   ccur1 24160
            16.11.12  Order theory (Extensible Structure Builder)   corhom 24173
            16.11.13  Order theory   cpresetrel 24181
            16.11.14  Finite composites ( i. e. finite sums, products ... )   cprd 24264
            16.11.15  Operation properties   ccm1 24297
            16.11.16  Groups and related structures   ridlideq 24301
            16.11.17  Free structures   csubsmg 24349
            16.11.18  Translations   trdom2 24357
            16.11.19  Fields and Rings   com2i 24382
            16.11.20  Ideals   cidln 24409
            16.11.21  Generic modules and vector spaces (New Structure builder)   cact 24413
            16.11.22  Generic modules and vector spaces   cvec 24415
            16.11.23  Real vector spaces   cvr 24455
            16.11.24  Matrices   cmmat 24459
            16.11.25  Affine spaces   craffsp 24465
            16.11.26  Intervals of reals and extended reals   bsi 24467
            16.11.27  Topology   topnem 24478
            16.11.28  Continuous functions   cnrsfin 24491
            16.11.29  Homeomorphisms   dmhmph 24499
            16.11.30  Initial and final topologies   intopcoaconlem3b 24504
            16.11.31  Filters   efilcp 24518
            16.11.32  Limits   plimfil 24524
            16.11.33  Uniform spaces   cunifsp 24551
            16.11.34  Separated spaces: T0, T1, T2 (Hausdorff) ...   hst1 24553
            16.11.35  Compactness   indcomp 24555
            16.11.36  Connectedness   singempcon 24559
            16.11.37  Topological fields   ctopfld 24563
            16.11.38  Standard topology on RR   intrn 24565
            16.11.39  Standard topology of intervals of RR   stoi 24567
            16.11.40  Cantor's set   cntrset 24568
            16.11.41  Pre-calculus and Cartesian geometry   dmse1 24569
            16.11.42  Extended Real numbers   nolimf 24585
            16.11.43  ( RR ^ N ) and ( CC ^ N )   cplcv 24610
            16.11.44  Calculus   cintvl 24662
            16.11.45  Directed multi graphs   cmgra 24674
            16.11.46  Category and deductive system underlying "structure"   calg 24677
            16.11.47  Deductive systems   cded 24700
            16.11.48  Categories   ccatOLD 24718
            16.11.49  Homsets   chomOLD 24751
            16.11.50  Monomorphisms, Epimorphisms, Isomorphisms   cepiOLD 24769
            16.11.51  Functors   cfuncOLD 24797
            16.11.52  Subcategories   csubcat 24809
            16.11.53  Terminal and initial objects   ciobj 24826
            16.11.54  Sources and sinks   csrce 24831
            16.11.55  Limits and co-limits   clmct 24840
            16.11.56  Product and sum of two objects   cprodo 24843
            16.11.57  Tarski's classes   ctar 24847
            16.11.58  Category Set   ccmrcase 24876
            16.11.59  Grammars, Logics, Machines and Automata   ckln 24946
            16.11.60  Words   cwrd 24947
            16.11.61  Planar geometry   cpoints 25022
      16.12  Mathbox for Jeff Hankins
            16.12.1  Miscellany   a1i13 25166
            16.12.2  Basic topological facts   topbnd 25208
            16.12.3  Topology of the real numbers   reconnOLD 25221
            16.12.4  Refinements   cfne 25225
            16.12.5  Neighborhood bases determine topologies   neibastop1 25274
            16.12.6  Lattice structure of topologies   topmtcl 25278
            16.12.7  Filter bases   fgmin 25285
            16.12.8  Directed sets, nets   tailfval 25287
      16.13  Mathbox for Jeff Madsen
            16.13.1  Logic and set theory   anim12da 25298
            16.13.2  Real and complex numbers; integers   fimaxreOLD 25396
            16.13.3  Sequences and sums   sdclem2 25418
            16.13.4  Topology   unopnOLD 25430
            16.13.5  Metric spaces   metf1o 25435
            16.13.6  Continuous maps and homeomorphisms   constcncf 25444
            16.13.7  Product topologies   txtopiOLD 25452
            16.13.8  Boundedness   ctotbnd 25456
            16.13.9  Isometries   cismty 25488
            16.13.10  Heine-Borel Theorem   heibor1lem 25499
            16.13.11  Banach Fixed Point Theorem   bfplem1 25512
            16.13.12  Euclidean space   crrn 25515
            16.13.13  Intervals (continued)   ismrer1 25528
            16.13.14  Groups and related structures   exidcl 25532
            16.13.15  Rings   rngonegcl 25542
            16.13.16  Ring homomorphisms   crnghom 25557
            16.13.17  Commutative rings   ccring 25586
            16.13.18  Ideals   cidl 25598
            16.13.19  Prime rings and integral domains   cprrng 25637
            16.13.20  Ideal generators   cigen 25650
      16.14  Mathbox for Rodolfo Medina
            16.14.1  Partitions   prtlem60 25669
      16.15  Mathbox for Stefan O'Rear
            16.15.1  Additional elementary logic and set theory   nelss 25717
            16.15.2  Additional theory of functions   fninfp 25720
            16.15.3  Extensions beyond function theory   gsumvsmul 25730
            16.15.4  Additional topology   elrfi 25735
            16.15.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 25739
            16.15.6  Algebraic closure systems   cnacs 25743
            16.15.7  Miscellanea 1. Map utilities   constmap 25754
            16.15.8  Miscellanea for polynomials   ofmpteq 25763
            16.15.9  Multivariate polynomials over the integers   cmzpcl 25765
            16.15.10  Miscellanea for Diophantine sets 1   coeq0 25797
            16.15.11  Diophantine sets 1: definitions   cdioph 25800
            16.15.12  Diophantine sets 2 miscellanea   ellz1 25812
            16.15.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 25818
            16.15.14  Diophantine sets 3: construction   diophrex 25821
            16.15.15  Diophantine sets 4 miscellanea   2sbcrex 25830
            16.15.16  Diophantine sets 4: Quantification   rexrabdioph 25841
            16.15.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 25848
            16.15.18  Diophantine sets 6 miscellanea   fz1ssnn 25858
            16.15.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 25860
            16.15.20  Pigeonhole Principle and cardinality helpers   fphpd 25865
            16.15.21  A non-closed set of reals is infinite   rencldnfilem 25869
            16.15.22  Miscellanea for Lagrange's theorem   icodiamlt 25871
            16.15.23  Lagrange's rational approximation theorem   irrapxlem1 25873
            16.15.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 25880
            16.15.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 25887
            16.15.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 25929
            16.15.27  Logarithm laws generalized to an arbitrary base   reglogcl 25941
            16.15.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 25949
            16.15.29  X and Y sequences 1: Definition and recurrence laws   crmx 25951
            16.15.30  Ordering and induction lemmas for the integers   monotuz 25992
            16.15.31  X and Y sequences 2: Order properties   rmxypos 26000
            16.15.32  Congruential equations   congtr 26018
            16.15.33  Alternating congruential equations   acongid 26028
            16.15.34  Additional theorems on integer divisibility   bezoutr 26038
            16.15.35  X and Y sequences 3: Divisibility properties   jm2.18 26047
            16.15.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26064
            16.15.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26074
            16.15.38  Uncategorized stuff not associated with a major project   setindtr 26083
            16.15.39  More equivalents of the Axiom of Choice   axac10 26092
            16.15.40  Finitely generated left modules   clfig 26131
            16.15.41  Noetherian left modules I   clnm 26139
            16.15.42  Addenda for structure powers   pwssplit0 26153
            16.15.43  Direct sum of left modules   cdsmm 26163
            16.15.44  Free modules   cfrlm 26178
            16.15.45  Every set admits a group structure iff choice   unxpwdom3 26222
            16.15.46  Independent sets and families   clindf 26240
            16.15.47  Characterization of free modules   lmimlbs 26272
            16.15.48  Noetherian rings and left modules II   clnr 26279
            16.15.49  Hilbert's Basis Theorem   cldgis 26291
            16.15.50  Additional material on polynomials [DEPRECATED]   cmnc 26301
            16.15.51  Degree and minimal polynomial of algebraic numbers   cdgraa 26311
            16.15.52  Algebraic integers I   citgo 26328
            16.15.53  Finite cardinality [SO]   en1uniel 26346
            16.15.54  Words in monoids and ordered group sum   issubmd 26349
            16.15.55  Transpositions in the symmetric group   cpmtr 26350
            16.15.56  The sign of a permutation   cpsgn 26380
            16.15.57  The matrix algebra   cmmul 26405
            16.15.58  The determinant   cmdat 26449
            16.15.59  Endomorphism algebra   cmend 26455
            16.15.60  Subfields   csdrg 26469
            16.15.61  Cyclic groups and order   idomrootle 26477
            16.15.62  Cyclotomic polynomials   ccytp 26487
            16.15.63  Miscellaneous topology   fgraphopab 26495
      16.16  Mathbox for Steve Rodriguez
            16.16.1  Miscellanea   iso0 26502
            16.16.2  Function operations   caofcan 26506
            16.16.3  Calculus   lhe4.4ex1a 26512
      16.17  Mathbox for Andrew Salmon
            16.17.1  Principia Mathematica * 10   pm10.12 26519
            16.17.2  Principia Mathematica * 11   2alanimi 26533
            16.17.3  Predicate Calculus   sbeqal1 26563
            16.17.4  Principia Mathematica * 13 and * 14   pm13.13a 26574
            16.17.5  Set Theory   elnev 26605
            16.17.6  Arithmetic   addcomgi 26628
            16.17.7  Geometry   cplusr 26629
      16.18  Mathbox for Jarvin Udandy
      16.19  Mathbox for David A. Wheeler
            16.19.1  Greater than, greater than or equal to.   cge-real 26676
            16.19.2  Hyperbolic trig functions   csinh 26686
            16.19.3  Reciprocal trig functions (sec, csc, cot)   csec 26697
            16.19.4  Identities for "if"   ifnmfalse 26719
            16.19.5  Not-member-of   AnelBC 26720
            16.19.6  Decimal point   cdp2 26721
            16.19.7  Signum (sgn or sign) function   csgn 26729
            16.19.8  Ceiling function   ccei 26739
            16.19.9  Miscellaneous   2m1e1 26743
      16.20  Mathbox for Alan Sare
            16.20.1  Conventional Metamath proofs, some derived from VD proofs   iidn3 26744
            16.20.2  What is Virtual Deduction?   wvd1 26818
            16.20.3  Virtual Deduction Theorems   df-vd1 26819
            16.20.4  Theorems proved using virtual deduction   trsspwALT 27023
            16.20.5  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 27053
            16.20.6  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 27120
            16.20.7  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 27124
            16.20.8  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 27131
            16.20.9  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 27134
      16.21  Mathbox for Jonathan Ben-Naim
            16.21.1  First order logic and set theory   bnj170 27153
            16.21.2  Well founded induction and recursion   bnj110 27320
            16.21.3  The existence of a minimal element in certain classes   bnj69 27470
            16.21.4  Well-founded induction   bnj1204 27472
            16.21.5  Well-founded recursion, part 1 of 3   bnj60 27522
            16.21.6  Well-founded recursion, part 2 of 3   bnj1500 27528
            16.21.7  Well-founded recursion, part 3 of 3   bnj1522 27532
      16.22  Mathbox for Norm Megill
            16.22.1  Obsolete experiments to study ax-12o   ax12-2 27533
            16.22.2  Miscellanea   cnaddcom 27591
            16.22.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 27594
            16.22.4  Functionals and kernels of a left vector space (or module)   clfn 27677
            16.22.5  Opposite rings and dual vector spaces   cld 27743
            16.22.6  Ortholattices and orthomodular lattices   cops 27792
            16.22.7  Atomic lattices with covering property   ccvr 27882
            16.22.8  Hilbert lattices   chlt 27970
            16.22.9  Projective geometries based on Hilbert lattices   clln 28110
            16.22.10  Construction of a vector space from a Hilbert lattice   cdlema1N 28410
            16.22.11  Construction of involution and inner product from a Hilbert lattice   clpoN 30100

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