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PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
1.1  Pre-logic
1.2  Propositional calculus
1.3  Other axiomatizations of classical propositional calculus
1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
1.6  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
1.7  Existential uniqueness
1.8  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
2.1  ZF Set Theory - start with the Axiom of Extensionality
2.2  ZF Set Theory - add the Axiom of Replacement
2.3  ZF Set Theory - add the Axiom of Power Sets
2.4  ZF Set Theory - add the Axiom of Union
2.5  ZF Set Theory - add the Axiom of Regularity
2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
3.2  ZFC Set Theory - add the Axiom of Choice
3.3  ZFC Axioms with no distinct variable requirements
3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
4.1  Inaccessibles
4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
5.1  Construction and axiomatization of real and complex numbers
5.2  Derive the basic properties from the field axioms
5.3  Real and complex numbers - basic operations
5.4  Integer sets
5.5  Order sets
5.6  Elementary integer functions
5.7  Elementary real and complex functions
5.8  Elementary limits and convergence
5.9  Elementary trigonometry
5.10  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
6.1  Elementary properties of divisibility
6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
7.1  Extensible structures
7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
8.1  Categories
8.2  Arrows (disjointified hom-sets)
8.3  Examples of categories
8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
9.1  Presets and directed sets using extensible structures
9.2  Posets and lattices using extensible structures
PART 10  BASIC ALGEBRAIC STRUCTURES
10.1  Monoids
10.2  Groups
10.3  Abelian groups
10.4  Rings
10.5  Division rings and fields
10.6  Left modules
10.7  Vector spaces
10.8  Ideals
10.9  Associative algebras
10.10  Abstract multivariate polynomials
10.11  The complex numbers as an algebraic extensible structure
10.12  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC TOPOLOGY
11.1  Topology
11.2  Filters and filter bases
11.3  Uniform Stuctures and Spaces
11.4  Metric spaces
11.5  Complex metric vector spaces
PART 12  BASIC REAL AND COMPLEX ANALYSIS
12.1  Continuity
12.2  Integrals
12.3  Derivatives
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
13.1  Polynomials
13.2  Sequences and series
13.3  Basic trigonometry
13.4  Basic number theory
PART 14  GRAPH THEORY
14.1  Undirected graphs - basics
14.2  Eulerian paths and the Konigsberg Bridge problem
PART 15  GUIDES AND MISCELLANEA
15.1  Guides (conventions, explanations, and examples)
15.2  Humor
15.3  (Future - to be reviewed and classified)
PART 16  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)
16.1  Additional material on group theory
16.2  Additional material on rings and fields
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
17.1  Complex vector spaces
17.2  Normed complex vector spaces
17.3  Operators on complex vector spaces
17.4  Inner product (pre-Hilbert) spaces
17.5  Complex Banach spaces
17.6  Complex Hilbert spaces
PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
18.1  Axiomatization of complex pre-Hilbert spaces
18.2  Inner product and norms
18.3  Cauchy sequences and completeness axiom
18.4  Subspaces and projections
18.5  Properties of Hilbert subspaces
18.6  Operators on Hilbert spaces
18.7  States on a Hilbert lattice and Godowski's equation
18.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
19.1  Mathboxes for user contributions
19.2  Mathbox for Stefan Allan
19.3  Mathbox for Thierry Arnoux
19.4  Mathbox for Mario Carneiro
19.5  Mathbox for Paul Chapman
19.6  Mathbox for Drahflow
19.7  Mathbox for Scott Fenton
19.8  Mathbox for Anthony Hart
19.9  Mathbox for Chen-Pang He
19.10  Mathbox for Jeff Hoffman
19.11  Mathbox for Wolf Lammen
19.12  Mathbox for Brendan Leahy
19.13  Mathbox for Jeff Hankins
19.14  Mathbox for Jeff Madsen
19.15  Mathbox for Rodolfo Medina
19.16  Mathbox for Stefan O'Rear
19.17  Mathbox for Steve Rodriguez
19.18  Mathbox for Andrew Salmon
19.19  Mathbox for Glauco Siliprandi
19.20  Mathbox for Saveliy Skresanov
19.21  Mathbox for Jarvin Udandy
19.22  Mathbox for Alexander van der Vekens
19.23  Mathbox for David A. Wheeler
19.24  Mathbox for Alan Sare
19.25  Mathbox for Jonathan Ben-Naim
19.26  Mathbox for Norm Megill

(* means the section header has a description)
*PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
*1.1  Pre-logic
1.1.1  Inferences for assisting proof development   dummylink 1
*1.2  Propositional calculus
1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
*1.2.2  The axioms of propositional calculus   ax-mp 5
*1.2.3  Logical implication   mp2 9
*1.2.4  Logical negation   con4d 100
*1.2.5  Logical equivalence   wb 178
*1.2.6  Logical disjunction and conjunction   wo 359
1.2.7  Miscellaneous theorems of propositional calculus   pm5.21nd 870
1.2.8  Abbreviated conjunction and disjunction of three wff's   w3o 936
1.2.9  Logical 'nand' (Sheffer stroke)   wnan 1297
1.2.10  Logical 'xor'   wxo 1314
1.2.11  True and false constants   wtru 1326
*1.2.12  Truth tables   truantru 1346
1.2.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1372
*1.2.14  Half-adders and full adders in propositional calculus   whad 1388
1.3  Other axiomatizations of classical propositional calculus
1.3.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1414
1.3.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1433
*1.3.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1444
1.3.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1450
1.3.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1469
1.3.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1473
1.3.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1488
1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1511
1.3.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1524
*1.3.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1543
*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
1.4.1  Universal quantifier; define "exists" and "not free"   wal 1550
1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1556
1.4.3  Axiom scheme ax-5 (Quantified Implication)   ax-5 1567
1.4.4  Axiom scheme ax-17 (Distinctness) - first use of \$d   ax-17 1627
1.4.5  Equality predicate; define substitution   cv 1652
1.4.6  Axiom scheme ax-9 (Existence)   ax-9 1668
1.4.7  Axiom scheme ax-8 (Equality)   ax-8 1689
1.4.8  Membership predicate   wcel 1727
1.4.9  Axiom scheme ax-13 (Left Equality for Binary Predicate)   ax-13 1729
1.4.10  Axiom scheme ax-14 (Right Equality for Binary Predicate)   ax-14 1731
*1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1733
*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
1.5.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1746
1.5.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1751
1.5.3  Axiom scheme ax-11 (Substitution)   ax-11 1763
1.5.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1953
*1.6  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
*1.6.1  Obsolete schemes ax-5o ax-4 ax-6o ax-9o ax-10o ax-10 ax-11o ax-12o ax-15 ax-16   ax-4 2218
*1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o   ax4 2228
*1.6.3  Legacy theorems using obsolete axioms   ax17o 2240
1.7  Existential uniqueness
1.8  Other axiomatizations related to classical predicate calculus
1.8.1  Predicate calculus with all distinct variables   ax-7d 2378
*1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2384
*1.8.3  Intuitionistic logic   axia1 2408
*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 2423
2.1.2  Class abstractions (a.k.a. class builders)   cab 2428
2.1.3  Class form not-free predicate   wnfc 2565
2.1.4  Negated equality and membership   wne 2605
2.1.4.1  Negated equality   neii 2609
2.1.4.2  Negated membership   neli 2703
2.1.5  Restricted quantification   wral 2711
2.1.6  The universal class   cvv 2962
*2.1.7  Conditional equality (experimental)   wcdeq 3150
2.1.8  Russell's Paradox   ru 3166
2.1.9  Proper substitution of classes for sets   wsbc 3167
2.1.10  Proper substitution of classes for sets into classes   csb 3267
2.1.11  Define basic set operations and relations   cdif 3303
2.1.12  Subclasses and subsets   df-ss 3320
2.1.13  The difference, union, and intersection of two classes   difeq1 3444
2.1.13.1  The difference of two classes   difeq1 3444
2.1.13.2  The union of two classes   elun 3474
2.1.13.3  The intersection of two classes   elin 3516
2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3556
2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdif2 3592
2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3606
2.1.14  The empty set   c0 3613
*2.1.15  "Weak deduction theorem" for set theory   cif 3763
2.1.16  Power classes   cpw 3823
2.1.17  Unordered and ordered pairs   csn 3838
2.1.18  The union of a class   cuni 4039
2.1.19  The intersection of a class   cint 4074
2.1.20  Indexed union and intersection   ciun 4117
2.1.21  Disjointness   wdisj 4207
2.1.22  Binary relations   wbr 4237
2.1.23  Ordered-pair class abstractions (class builders)   copab 4290
2.1.24  Transitive classes   wtr 4327
2.2  ZF Set Theory - add the Axiom of Replacement
2.2.1  Introduce the Axiom of Replacement   ax-rep 4345
2.2.2  Derive the Axiom of Separation   axsep 4354
2.2.3  Derive the Null Set Axiom   zfnuleu 4360
2.2.4  Theorems requiring subset and intersection existence   nalset 4369
2.2.5  Theorems requiring empty set existence   class2set 4396
2.3  ZF Set Theory - add the Axiom of Power Sets
2.3.1  Introduce the Axiom of Power Sets   ax-pow 4406
2.3.2  Derive the Axiom of Pairing   zfpair 4430
2.3.3  Ordered pair theorem   opnz 4461
2.3.4  Ordered-pair class abstractions (cont.)   opabid 4490
2.3.5  Power class of union and intersection   pwin 4516
2.3.6  Epsilon and identity relations   cep 4521
2.3.7  Partial and complete ordering   wpo 4530
2.3.8  Founded and well-ordering relations   wfr 4567
2.3.9  Ordinals   word 4609
2.4  ZF Set Theory - add the Axiom of Union
2.4.1  Introduce the Axiom of Union   ax-un 4730
2.4.2  Ordinals (continued)   ordon 4792
2.4.3  Transfinite induction   tfi 4862
2.4.4  The natural numbers (i.e. finite ordinals)   com 4874
2.4.5  Peano's postulates   peano1 4893
2.4.6  Finite induction (for finite ordinals)   find 4899
2.4.7  Relations   cxp 4905
2.4.8  Definite description binder (inverted iota)   cio 5445
2.4.9  Functions   wfun 5477
2.4.10  Operations   co 6110
2.4.11  "Maps to" notation   elmpt2cl 6317
2.4.12  Function operation   cof 6332
2.4.13  First and second members of an ordered pair   c1st 6376
*2.4.14  Special "Maps to" operations   mpt2xopn0yelv 6493
2.4.15  Function transposition   ctpos 6507
2.4.16  Curry and uncurry   ccur 6546
2.4.17  Proper subset relation   crpss 6550
2.4.18  Iota properties   fvopab5 6563
2.4.19  Cantor's Theorem   canth 6568
2.4.20  Undefined values and restricted iota (description binder)   cund 6570
2.4.21  Functions on ordinals; strictly monotone ordinal functions   iunon 6629
2.4.22  "Strong" transfinite recursion   crecs 6661
2.4.23  Recursive definition generator   crdg 6696
2.4.24  Finite recursion   frfnom 6721
2.4.25  Abian's "most fundamental" fixed point theorem   abianfplem 6744
2.4.26  Ordinal arithmetic   c1o 6746
2.4.27  Natural number arithmetic   nna0 6876
2.4.28  Equivalence relations and classes   wer 6931
2.4.29  The mapping operation   cmap 7047
2.4.30  Infinite Cartesian products   cixp 7092
2.4.31  Equinumerosity   cen 7135
2.4.32  Schroeder-Bernstein Theorem   sbthlem1 7246
2.4.33  Equinumerosity (cont.)   xpf1o 7298
2.4.34  Pigeonhole Principle   phplem1 7315
2.4.35  Finite sets   onomeneq 7325
2.4.36  Finite intersections   cfi 7444
2.4.37  Hall's marriage theorem   marypha1lem 7467
2.4.38  Supremum   csup 7474
2.4.39  Ordinal isomorphism, Hartog's theorem   coi 7507
2.4.40  Hartogs function, order types, weak dominance   char 7553
2.5  ZF Set Theory - add the Axiom of Regularity
2.5.1  Introduce the Axiom of Regularity   ax-reg 7589
2.5.2  Axiom of Infinity equivalents   inf0 7605
2.6  ZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity   ax-inf 7622
2.6.2  Existence of omega (the set of natural numbers)   omex 7627
2.6.3  Cantor normal form   ccnf 7645
2.6.4  Transitive closure   trcl 7693
2.6.5  Rank   cr1 7717
2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7840
2.6.7  Cardinal numbers   ccrd 7853
2.6.8  Axiom of Choice equivalents   wac 8027
2.6.9  Cardinal number arithmetic   ccda 8078
2.6.10  The Ackermann bijection   ackbij2lem1 8130
2.6.11  Cofinality (without Axiom of Choice)   cflem 8157
2.6.12  Eight inequivalent definitions of finite set   sornom 8188
2.6.13  Hereditarily size-limited sets without Choice   itunifval 8327
*PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
3.1.1  Introduce the Axiom of Countable Choice   ax-cc 8346
3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 8357
3.2  ZFC Set Theory - add the Axiom of Choice
3.2.1  Introduce the Axiom of Choice   ax-ac 8370
3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8405
3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8452
3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8480
3.2.5  Cofinality using Axiom of Choice   alephreg 8488
3.3  ZFC Axioms with no distinct variable requirements
3.4  The Generalized Continuum Hypothesis
3.4.1  Sets satisfying the Generalized Continuum Hypothesis   cgch 8526
3.4.2  Derivation of the Axiom of Choice   gchaclem 8584
*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
4.1  Inaccessibles
4.1.1  Weakly and strongly inaccessible cardinals   cwina 8588
4.1.2  Weak universes   cwun 8606
4.1.3  Tarski's classes   ctsk 8654
4.1.4  Grothendieck's universes   cgru 8696
4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8729
4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8732
4.2.3  Tarski map function   ctskm 8743
*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 8750
5.1.2  Final derivation of real and complex number postulates   axaddf 9051
5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9077
5.2  Derive the basic properties from the field axioms
5.2.1  Some deductions from the field axioms for complex numbers   cnex 9102
5.2.2  Infinity and the extended real number system   cpnf 9148
5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9178
5.2.4  Ordering on reals   lttr 9183
5.2.5  Initial properties of the complex numbers   mul12 9263
5.3  Real and complex numbers - basic operations
5.3.2  Subtraction   cmin 9322
5.3.3  Multiplication   muladd 9497
5.3.4  Ordering on reals (cont.)   gt0ne0 9524
5.3.5  Reciprocals   ixi 9682
5.3.6  Division   cdiv 9708
5.3.7  Ordering on reals (cont.)   elimgt0 9877
5.3.8  Completeness Axiom and Suprema   fimaxre 9986
5.3.9  Imaginary and complex number properties   inelr 10021
5.3.10  Function operation analogue theorems   ofsubeq0 10028
5.4  Integer sets
5.4.1  Natural numbers (as a subset of complex numbers)   cn 10031
5.4.2  Principle of mathematical induction   nnind 10049
*5.4.3  Decimal representation of numbers   c2 10080
*5.4.4  Some properties of specific numbers   0p1e1 10124
5.4.5  The Archimedean property   nnunb 10248
5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 10252
5.4.7  Integers (as a subset of complex numbers)   cz 10313
5.4.8  Decimal arithmetic   cdc 10413
5.4.9  Upper partititions of integers   cuz 10519
5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10600
5.4.11  Rational numbers (as a subset of complex numbers)   cq 10605
5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10631
5.5  Order sets
5.5.1  Positive reals (as a subset of complex numbers)   crp 10643
5.5.2  Infinity and the extended real number system (cont.)   cxne 10738
5.5.3  Supremum on the extended reals   xrsupexmnf 10914
5.5.4  Real number intervals   cioo 10947
5.5.5  Finite intervals of integers   cfz 11074
5.5.6  Half-open integer ranges   cfzo 11166
5.6  Elementary integer functions
5.6.1  The floor (greatest integer) function   cfl 11232
5.6.2  The modulo (remainder) operation   cmo 11281
5.6.3  The infinite sequence builder "seq"   om2uz0i 11318
5.6.4  Integer powers   cexp 11413
5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11591
5.6.6  Factorial function   cfa 11597
5.6.7  The binomial coefficient operation   cbc 11624
5.6.8  The ` # ` (finite set size) function   chash 11649
5.6.8.1  Finite induction on the size of the first component of a binary relation   brfi1indlem 11745
5.6.9  Words over a set   cword 11748
5.6.10  Longer string literals   cs2 11836
5.7  Elementary real and complex functions
5.7.1  The "shift" operation   cshi 11912
5.7.2  Real and imaginary parts; conjugate   ccj 11932
5.7.3  Square root; absolute value   csqr 12069
5.8  Elementary limits and convergence
5.8.1  Superior limit (lim sup)   clsp 12295
5.8.2  Limits   cli 12309
5.8.3  Finite and infinite sums   csu 12510
5.8.4  The binomial theorem   binomlem 12639
5.8.5  The inclusion/exclusion principle   incexclem 12647
5.8.6  Infinite sums (cont.)   isumshft 12650
5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12663
5.8.8  Arithmetic series   arisum 12670
5.8.9  Geometric series   expcnv 12674
5.8.10  Ratio test for infinite series convergence   cvgrat 12691
5.8.11  Mertens' theorem   mertenslem1 12692
5.9  Elementary trigonometry
5.9.1  The exponential, sine, and cosine functions   ce 12695
5.9.2  _e is irrational   eirrlem 12834
5.10  Cardinality of real and complex number subsets
5.10.1  Countability of integers and rationals   xpnnen 12839
5.10.2  The reals are uncountable   rpnnen2lem1 12845
*PART 6  ELEMENTARY NUMBER THEORY
6.1  Elementary properties of divisibility
6.1.1  Irrationality of square root of 2   sqr2irrlem 12878
6.1.2  Some Number sets are chains of proper subsets   nthruc 12881
6.1.3  The divides relation   cdivides 12883
6.1.4  The division algorithm   divalglem0 12944
6.1.5  Bit sequences   cbits 12962
6.1.6  The greatest common divisor operator   cgcd 13037
6.1.7  Bézout's identity   bezoutlem1 13069
6.1.8  Algorithms   nn0seqcvgd 13092
6.1.9  Euclid's Algorithm   eucalgval2 13103
6.2  Elementary prime number theory
6.2.1  Elementary properties   cprime 13110
6.2.2  Properties of the canonical representation of a rational   cnumer 13156
6.2.3  Euler's theorem   codz 13183
6.2.4  Pythagorean Triples   coprimeprodsq 13214
6.2.5  The prime count function   cpc 13241
6.2.6  Pocklington's theorem   prmpwdvds 13303
6.2.7  Infinite primes theorem   unbenlem 13307
6.2.8  Sum of prime reciprocals   prmreclem1 13315
6.2.9  Fundamental theorem of arithmetic   1arithlem1 13322
6.2.10  Lagrange's four-square theorem   cgz 13328
6.2.11  Van der Waerden's theorem   cvdwa 13364
6.2.12  Ramsey's theorem   cram 13398
6.2.13  Decimal arithmetic (cont.)   dec2dvds 13430
6.2.14  Specific prime numbers   4nprm 13458
6.2.15  Very large primes   1259lem1 13481
PART 7  BASIC STRUCTURES
7.1  Extensible structures
*7.1.1  Basic definitions   cstr 13496
7.1.2  Slot definitions   cplusg 13560
7.1.3  Definition of the structure product   crest 13679
7.1.4  Definition of the structure quotient   cordt 13752
7.2  Moore spaces
7.2.1  Moore closures   mrcflem 13862
7.2.2  Independent sets in a Moore system   mrisval 13886
7.2.3  Algebraic closure systems   isacs 13907
PART 8  BASIC CATEGORY THEORY
8.1  Categories
8.1.1  Categories   ccat 13920
8.1.2  Opposite category   coppc 13968
8.1.3  Monomorphisms and epimorphisms   cmon 13985
8.1.4  Sections, inverses, isomorphisms   csect 14001
8.1.5  Subcategories   cssc 14038
8.1.6  Functors   cfunc 14082
8.1.7  Full & faithful functors   cful 14130
8.1.8  Natural transformations and the functor category   cnat 14169
8.2  Arrows (disjointified hom-sets)
8.2.1  Identity and composition for arrows   cida 14239
8.3  Examples of categories
8.3.1  The category of sets   csetc 14261
8.3.2  The category of categories   ccatc 14280
8.4  Categorical constructions
8.4.1  Product of categories   cxpc 14296
8.4.2  Functor evaluation   cevlf 14337
8.4.3  Hom functor   chof 14376
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 14428
9.2.2  Lattices   clat 14505
9.2.3  The dual of an ordered set   codu 14586
9.2.4  Subset order structures   cipo 14608
9.2.5  Distributive lattices   latmass 14645
9.2.6  Posets and lattices as relations   cps 14655
9.2.7  Directed sets, nets   cdir 14704
PART 10  BASIC ALGEBRAIC STRUCTURES
10.1  Monoids
10.1.1  Definition and basic properties   cmnd 14715
10.1.2  Monoid homomorphisms and submonoids   cmhm 14767
*10.1.3  Ordered group sum operation   gsumvallem1 14802
10.1.4  Free monoids   cfrmd 14823
10.2  Groups
10.2.1  Definition and basic properties   df-grp 14843
10.2.2  Subgroups and Quotient groups   csubg 14969
10.2.3  Elementary theory of group homomorphisms   cghm 15034
10.2.4  Isomorphisms of groups   cgim 15075
10.2.5  Group actions   cga 15097
10.2.6  Symmetry groups and Cayley's Theorem   csymg 15123
10.2.7  Centralizers and centers   ccntz 15145
10.2.8  The opposite group   coppg 15172
10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 15194
10.2.10  Direct products   clsm 15299
10.2.11  Free groups   cefg 15369
10.3  Abelian groups
10.3.1  Definition and basic properties   ccmn 15443
10.3.2  Cyclic groups   ccyg 15518
10.3.3  Group sum operation   gsumval3a 15543
10.3.4  Internal direct products   cdprd 15585
10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15654
10.4  Rings
10.4.1  Multiplicative Group   cmgp 15679
10.4.2  Definition and basic properties   crg 15691
10.4.3  Opposite ring   coppr 15758
10.4.4  Divisibility   cdsr 15774
10.4.5  Ring homomorphisms   crh 15848
10.5  Division rings and fields
10.5.1  Definition and basic properties   cdr 15866
10.5.2  Subrings of a ring   csubrg 15895
10.5.3  Absolute value (abstract algebra)   cabv 15935
10.5.4  Star rings   cstf 15962
10.6  Left modules
10.6.1  Definition and basic properties   clmod 15981
10.6.2  Subspaces and spans in a left module   clss 16039
10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 16126
10.6.4  Subspace sum; bases for a left module   clbs 16177
10.7  Vector spaces
10.7.1  Definition and basic properties   clvec 16205
10.8  Ideals
10.8.1  The subring algebra; ideals   csra 16271
10.8.2  Two-sided ideals and quotient rings   c2idl 16333
10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 16343
10.8.4  Nonzero rings   cnzr 16359
10.8.5  Left regular elements. More kinds of rings   crlreg 16370
10.9  Associative algebras
10.9.1  Definition and basic properties   casa 16400
10.10  Abstract multivariate polynomials
10.10.1  Definition and basic properties   cmps 16437
10.10.2  Polynomial evaluation   evlslem4 16595
10.10.3  Univariate polynomials   cps1 16600
10.11  The complex numbers as an algebraic extensible structure
10.11.1  Definition and basic properties   cpsmet 16716
10.11.2  Algebraic constructions based on the complexes   czrh 16809
10.12  Generalized pre-Hilbert and Hilbert spaces
10.12.1  Definition and basic properties   cphl 16886
10.12.2  Orthocomplements and closed subspaces   cocv 16918
10.12.3  Orthogonal projection and orthonormal bases   cpj 16958
PART 11  BASIC TOPOLOGY
11.1  Topology
11.1.1  Topological spaces   ctop 16989
11.1.2  TopBases for topologies   isbasisg 17043
11.1.3  Examples of topologies   distop 17091
11.1.4  Closure and interior   ccld 17111
11.1.5  Neighborhoods   cnei 17192
11.1.6  Limit points and perfect sets   clp 17229
11.1.7  Subspace topologies   restrcl 17252
11.1.8  Order topology   ordtbaslem 17283
11.1.9  Limits and continuity in topological spaces   ccn 17319
11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17401
11.1.11  Compactness   ccmp 17480
11.1.12  Bolzano-Weierstrass theorem   bwth 17504
11.1.13  Connectedness   ccon 17505
11.1.14  First- and second-countability   c1stc 17531
11.1.15  Local topological properties   clly 17558
11.1.16  Compactly generated spaces   ckgen 17596
11.1.17  Product topologies   ctx 17623
11.1.18  Continuous function-builders   cnmptid 17724
11.1.19  Quotient maps and quotient topology   ckq 17756
11.1.20  Homeomorphisms   chmeo 17816
11.2  Filters and filter bases
11.2.1  Filter bases   elmptrab 17890
11.2.2  Filters   cfil 17908
11.2.3  Ultrafilters   cufil 17962
11.2.4  Filter limits   cfm 17996
11.2.5  Extension by continuity   ccnext 18121
11.2.6  Topological groups   ctmd 18131
11.2.7  Infinite group sum on topological groups   ctsu 18186
11.2.8  Topological rings, fields, vector spaces   ctrg 18216
11.3  Uniform Stuctures and Spaces
11.3.1  Uniform structures   cust 18260
11.3.2  The topology induced by an uniform structure   cutop 18291
11.3.3  Uniform Spaces   cuss 18314
11.3.4  Uniform continuity   cucn 18336
11.3.5  Cauchy filters in uniform spaces   ccfilu 18347
11.3.6  Complete uniform spaces   ccusp 18358
11.4  Metric spaces
11.4.1  Pseudometric spaces   ispsmet 18366
11.4.2  Basic metric space properties   cxme 18378
11.4.3  Metric space balls   blfvalps 18444
11.4.4  Open sets of a metric space   mopnval 18499
11.4.5  Continuity in metric spaces   metcnp3 18601
11.4.6  The uniform structure generated by a metric   metuvalOLD 18610
11.4.7  Examples of metric spaces   dscmet 18651
11.4.8  Normed algebraic structures   cnm 18655
11.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 18770
11.4.10  Topology on the reals   qtopbaslem 18823
11.4.11  Topological definitions using the reals   cii 18936
11.4.12  Path homotopy   chtpy 19023
11.4.13  The fundamental group   cpco 19056
11.5  Complex metric vector spaces
11.5.1  Complex left modules   cclm 19118
11.5.2  Complex pre-Hilbert space   ccph 19160
11.5.3  Convergence and completeness   ccfil 19236
11.5.4  Baire's Category Theorem   bcthlem1 19308
11.5.5  Banach spaces and complex Hilbert spaces   ccms 19316
11.5.6  Minimizing Vector Theorem   minveclem1 19356
11.5.7  Projection Theorem   pjthlem1 19369
PART 12  BASIC REAL AND COMPLEX ANALYSIS
12.1  Continuity
12.1.1  Intermediate value theorem   pmltpclem1 19376
12.2  Integrals
12.2.1  Lebesgue measure   covol 19390
12.2.2  Lebesgue integration   cmbf 19537
12.2.2.1  Lesbesgue integral   cmbf 19537
12.2.2.2  Lesbesgue directed integral   cdit 19764
12.3  Derivatives
12.3.1  Real and complex differentiation   climc 19780
12.3.1.1  Derivatives of functions of one complex or real variable   climc 19780
12.3.1.2  Results on real differentiation   dvferm1lem 19899
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
13.1  Polynomials
13.1.1  Abstract polynomials, continued   evlslem6 19965
13.1.2  Polynomial degrees   cmdg 20007
13.1.3  The division algorithm for univariate polynomials   cmn1 20079
13.1.4  Elementary properties of complex polynomials   cply 20134
13.1.5  The division algorithm for polynomials   cquot 20238
13.1.6  Algebraic numbers   caa 20262
13.1.7  Liouville's approximation theorem   aalioulem1 20280
13.2  Sequences and series
13.2.1  Taylor polynomials and Taylor's theorem   ctayl 20300
13.2.2  Uniform convergence   culm 20323
13.2.3  Power series   pserval 20357
13.3  Basic trigonometry
13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 20390
13.3.2  Properties of pi = 3.14159...   pilem1 20398
13.3.3  Mapping of the exponential function   efgh 20474
13.3.4  The natural logarithm on complex numbers   clog 20483
13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20674
13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20710
13.3.7  Inverse trigonometric functions   casin 20733
13.3.8  The Birthday Problem   log2ublem1 20817
13.3.9  Areas in R^2   carea 20825
13.3.10  More miscellaneous converging sequences   rlimcnp 20835
13.3.11  Inequality of arithmetic and geometric means   cvxcl 20854
13.3.12  Euler-Mascheroni constant   cem 20861
13.4  Basic number theory
13.4.1  Wilson's theorem   wilthlem1 20882
13.4.2  The Fundamental Theorem of Algebra   ftalem1 20886
13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20894
13.4.4  Number-theoretical functions   ccht 20904
13.4.5  Perfect Number Theorem   mersenne 21042
13.4.6  Characters of Z/nZ   cdchr 21047
13.4.7  Bertrand's postulate   bcctr 21090
13.4.8  Legendre symbol   clgs 21109
13.4.9  Quadratic reciprocity   lgseisenlem1 21164
13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 21178
13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 21194
13.4.12  The Prime Number Theorem   mudivsum 21255
13.4.13  Ostrowski's theorem   abvcxp 21340
*PART 14  GRAPH THEORY
14.1  Undirected graphs - basics
14.1.1  Undirected hypergraphs   cuhg 21365
14.1.2  Undirected multigraphs   cumg 21378
14.1.3  Undirected simple graphs   cuslg 21395
14.1.3.1  Undirected simple graphs - basics   cuslg 21395
14.1.3.2  Undirected simple graphs - examples   usgraexvlem 21445
14.1.3.3  Finite undirected simple graphs   fiusgraedgfi 21452
14.1.4  Neighbors, complete graphs and universal vertices   cnbgra 21461
14.1.4.1  Neighbors   nbgraop 21467
14.1.4.2  Complete graphs   iscusgra 21496
14.1.4.3  Universal vertices   isuvtx 21528
14.1.5  Walks, paths and cycles   cwalk 21537
14.1.5.1  Walks and trails   wlks 21557
14.1.5.2  Paths and simple paths   pths 21597
14.1.5.3  Circuits and cycles   crcts 21640
14.1.5.4  Connected graphs   cconngra 21687
14.1.6  Vertex Degree   cvdg 21695
14.2  Eulerian paths and the Konigsberg Bridge problem
14.2.1  Eulerian paths   ceup 21715
14.2.2  The Konigsberg Bridge problem   vdeg0i 21735
PART 15  GUIDES AND MISCELLANEA
15.1  Guides (conventions, explanations, and examples)
*15.1.1  Conventions   conventions 21741
15.1.2  Natural deduction   natded 21742
*15.1.3  Natural deduction examples   ex-natded5.2 21743
15.1.4  Definitional examples   ex-or 21760
15.2  Humor
15.2.1  April Fool's theorem   avril1 21788
15.3  (Future - to be reviewed and classified)
15.3.1  Planar incidence geometry   cplig 21794
15.3.2  Algebra preliminaries   crpm 21799
15.3.3  Transitive closure   ctcl 21801
*PART 16  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)
16.1  Additional material on group theory
16.1.1  Definitions and basic properties for groups   cgr 21805
16.1.2  Definition and basic properties of Abelian groups   cablo 21900
16.1.3  Subgroups   csubgo 21920
16.1.4  Operation properties   cass 21931
16.1.5  Group-like structures   cmagm 21937
16.1.6  Examples of Abelian groups   ablosn 21966
16.1.7  Group homomorphism and isomorphism   cghom 21976
16.2  Additional material on rings and fields
16.2.1  Definition and basic properties   crngo 21994
16.2.2  Examples of rings   cnrngo 22022
16.2.3  Division Rings   cdrng 22024
16.2.4  Star Fields   csfld 22027
16.2.5  Fields and Rings   ccm2 22029
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
17.1  Complex vector spaces
17.1.1  Definition and basic properties   cvc 22055
17.1.2  Examples of complex vector spaces   cncvc 22093
17.2  Normed complex vector spaces
17.2.1  Definition and basic properties   cnv 22094
17.2.2  Examples of normed complex vector spaces   cnnv 22199
17.2.3  Induced metric of a normed complex vector space   imsval 22208
17.2.4  Inner product   cdip 22227
17.2.5  Subspaces   css 22251
17.3  Operators on complex vector spaces
17.3.1  Definitions and basic properties   clno 22272
17.4  Inner product (pre-Hilbert) spaces
17.4.1  Definition and basic properties   ccphlo 22344
17.4.2  Examples of pre-Hilbert spaces   cncph 22351
17.4.3  Properties of pre-Hilbert spaces   isph 22354
17.5  Complex Banach spaces
17.5.1  Definition and basic properties   ccbn 22395
17.5.2  Examples of complex Banach spaces   cnbn 22402
17.5.3  Uniform Boundedness Theorem   ubthlem1 22403
17.5.4  Minimizing Vector Theorem   minvecolem1 22407
17.6  Complex Hilbert spaces
17.6.1  Definition and basic properties   chlo 22418
17.6.2  Standard axioms for a complex Hilbert space   hlex 22431
17.6.3  Examples of complex Hilbert spaces   cnchl 22449
17.6.4  Subspaces   ssphl 22450
17.6.5  Hellinger-Toeplitz Theorem   htthlem 22451
*PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
18.1  Axiomatization of complex pre-Hilbert spaces
18.1.1  Basic Hilbert space definitions   chil 22453
18.1.2  Preliminary ZFC lemmas   df-hnorm 22502
*18.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 22515
*18.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 22533
18.1.5  Vector operations   hvmulex 22545
18.1.6  Inner product postulates for a Hilbert space   ax-hfi 22612
18.2  Inner product and norms
18.2.1  Inner product   his5 22619
18.2.2  Norms   dfhnorm2 22655
18.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 22693
18.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 22712
18.3  Cauchy sequences and completeness axiom
18.3.1  Cauchy sequences and limits   hcau 22717
18.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 22727
18.3.3  Completeness postulate for a Hilbert space   ax-hcompl 22735
18.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 22736
18.4  Subspaces and projections
18.4.1  Subspaces   df-sh 22740
18.4.2  Closed subspaces   df-ch 22755
18.4.3  Orthocomplements   df-oc 22785
18.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 22841
18.4.5  Projection theorem   pjhthlem1 22924
18.4.6  Projectors   df-pjh 22928
18.5  Properties of Hilbert subspaces
18.5.1  Orthomodular law   omlsilem 22935
18.5.2  Projectors (cont.)   pjhtheu2 22949
18.5.3  Hilbert lattice operations   sh0le 22973
18.5.4  Span (cont.) and one-dimensional subspaces   spansn0 23074
18.5.5  Commutes relation for Hilbert lattice elements   df-cm 23116
18.5.6  Foulis-Holland theorem   fh1 23151
18.5.7  Quantum Logic Explorer axioms   qlax1i 23160
18.5.8  Orthogonal subspaces   chscllem1 23170
18.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 23187
18.5.10  Projectors (cont.)   pjorthi 23202
18.5.11  Mayet's equation E_3   mayete3i 23261
18.6  Operators on Hilbert spaces
*18.6.1  Operator sum, difference, and scalar multiplication   df-hosum 23264
18.6.2  Zero and identity operators   df-h0op 23282
18.6.3  Operations on Hilbert space operators   hoaddcl 23292
18.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 23373
18.6.5  Linear and continuous functionals and norms   df-nmfn 23379
18.6.7  Dirac bra-ket notation   df-bra 23384
18.6.8  Positive operators   df-leop 23386
18.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 23387
18.6.10  Theorems about operators and functionals   nmopval 23390
18.6.11  Riesz lemma   riesz3i 23596
18.6.13  Quantum computation error bound theorem   unierri 23638
18.6.14  Dirac bra-ket notation (cont.)   branmfn 23639
18.6.15  Positive operators (cont.)   leopg 23656
18.6.16  Projectors as operators   pjhmopi 23680
18.7  States on a Hilbert lattice and Godowski's equation
18.7.1  States on a Hilbert lattice   df-st 23745
18.7.2  Godowski's equation   golem1 23805
18.8  Cover relation, atoms, exchange axiom, and modular symmetry
18.8.1  Covers relation; modular pairs   df-cv 23813
18.8.2  Atoms   df-at 23872
18.8.3  Superposition principle   superpos 23888
18.8.4  Atoms, exchange and covering properties, atomicity   chcv1 23889
18.8.5  Irreducibility   chirredlem1 23924
18.8.6  Atoms (cont.)   atcvat3i 23930
18.8.7  Modular symmetry   mdsymlem1 23937
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
19.1  Mathboxes for user contributions
19.1.1  Mathbox guidelines   mathbox 23976
19.2  Mathbox for Stefan Allan
19.3  Mathbox for Thierry Arnoux
19.3.1  Propositional Calculus - misc additions   bian1d 23981
19.3.2  Predicate Calculus   abeq2f 23991
19.3.2.1  Predicate Calculus - misc additions   abeq2f 23991
19.3.2.2  Restricted quantification - misc additions   reximddv 23993
19.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 24002
19.3.2.4  Existential "at most one" - misc additions   mo5f 24003
19.3.2.5  Existential uniqueness - misc additions   2reuswap2 24006
19.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 24010
19.3.3  General Set Theory   ceqsexv2d 24016
19.3.3.1  Class abstractions (a.k.a. class builders)   ceqsexv2d 24016
19.3.3.2  Image Sets   abrexdomjm 24019
19.3.3.3  Set relations and operations - misc additions   eqri 24025
19.3.3.4  Unordered pairs   elpreq 24030
19.3.3.5  Conditional operator - misc additions   ifeqeqx 24032
19.3.3.6  Indexed union - misc additions   iuneq12daf 24038
19.3.3.7  Disjointness - misc additions   cbvdisjf 24046
19.3.4  Relations and Functions   dfrel4 24065
19.3.4.1  Relations - misc additions   dfrel4 24065
19.3.4.2  Functions - misc additions   fdmrn 24070
19.3.4.3  Isomorphisms - misc. add.   gtiso 24119
19.3.4.4  Disjointness (additional proof requiring functions)   disjdsct 24121
19.3.4.5  First and second members of an ordered pair - misc additions   df1stres 24122
19.3.4.6  Supremum - misc additions   supssd 24129
19.3.4.7  Countable Sets   nnct 24130
19.3.5  Real and Complex Numbers   addeq0 24145
19.3.5.2  Ordering on reals - misc additions   lt2addrd 24146
19.3.5.3  Extended reals - misc additions   xgepnf 24147
19.3.5.4  Real number intervals - misc additions   icossicc 24160
19.3.5.5  Finite intervals of integers - misc additions   fzssnn 24178
19.3.5.6  Half-open integer ranges - misc additions   iundisjfi 24183
19.3.5.7  The ` # ` (finite set size) function - misc additions   hashresfn 24187
19.3.5.8  The greatest common divisor operator - misc. add   numdenneg 24191
19.3.5.9  Integers   ltesubnnd 24193
19.3.5.10  Division in the extended real number system   cxdiv 24194
19.3.6  Structure builders   ress0g 24213
19.3.6.1  Structure builder restriction operator   ress0g 24213
19.3.6.2  Posets   tospos 24217
19.3.6.3  Complete lattices   clatp0ex 24224
19.3.6.4  Extended reals Structure - misc additions   ax-xrssca 24226
19.3.6.5  The extended non-negative real numbers monoid   xrge0base 24238
19.3.7  Algebra   sumpr 24249
19.3.7.1  Finitely supported group sums - misc additions   sumpr 24249
19.3.7.2  Rings - misc additions   dvrdir 24257
19.3.7.3  Ordered groups   cogrp 24262
19.3.7.4  Ordered fields   cofld 24264
19.3.7.5  The Archimedean property for generic algebraic structures   cinftm 24277
19.3.7.6  Ring homomorphisms - misc additions   rhmdvdsr 24287
19.3.7.7  The ring of integers   zzsbase 24294
19.3.7.8  The ordered field of reals   rebase 24300
19.3.8  Topology   cmetid 24312
19.3.8.1  Pseudometrics   cmetid 24312
19.3.8.2  Continuity - misc additions   hauseqcn 24324
19.3.8.3  Topology of the closed unit   unitsscn 24325
19.3.8.4  Topology of ` ( RR X. RR ) `   unicls 24332
19.3.8.5  Order topology - misc. additions   cnvordtrestixx 24342
19.3.8.6  Continuity in topological spaces - misc. additions   mndpluscn 24343
19.3.8.7  Topology of the extended non-negative real numbers monoid   xrge0hmph 24349
19.3.8.8  Limits - misc additions   lmlim 24364
19.3.9  Uniform Stuctures and Spaces   chcmp 24371
19.3.9.1  Hausdorff Completion   chcmp 24371
19.3.10  Topology and algebraic structures   zzsnm 24373
19.3.10.1  The norm on the ring of the integer numbers   zzsnm 24373
19.3.10.2  The complete ordered field of the real numbers   recms 24374
19.3.10.3  Topological ` ZZ ` -modules   zlm0 24377
19.3.10.4  The canonical embedding of the rational numbers into a division ring   cqqh 24387
19.3.10.5  The canonical embedding of ` RR ` into a complete ordered field   crrh 24408
19.3.10.6  Embedding into ` RR* `   cxrh 24413
19.3.10.7  Canonical embeddings into ` RR `   zrhre 24416
19.3.11  Real and complex functions   clogb 24419
*19.3.11.1  Logarithm laws generalized to an arbitrary base - logb   clogb 24419
19.3.11.2  Indicator Functions   cind 24439
19.3.11.3  Extended sum   cesum 24455
19.3.12  Mixed Function/Constant operation   cofc 24509
19.3.13  Abstract measure   csiga 24521
19.3.13.1  Sigma-Algebra   csiga 24521
19.3.13.2  Generated Sigma-Algebra   csigagen 24552
19.3.13.3  The Borel algebra on the real numbers   cbrsiga 24566
19.3.13.4  Product Sigma-Algebra   csx 24573
19.3.13.5  Measures   cmeas 24580
19.3.13.6  The counting measure   cntmeas 24611
19.3.13.7  The Lebesgue measure - misc additions   volss 24614
19.3.13.8  The 'almost everywhere' relation   cae 24619
19.3.13.9  Measurable functions   cmbfm 24631
19.3.13.10  Borel Algebra on ` ( RR X. RR ) `   br2base 24650
19.3.14  Integration   itgeq12dv 24672
19.3.14.1  Lebesgue integral - misc additions   itgeq12dv 24672
19.3.14.2  Bochner integral   citgm 24673
19.3.15  Probability   cprb 24696
19.3.15.1  Probability Theory   cprb 24696
19.3.15.2  Conditional Probabilities   ccprob 24720
19.3.15.3  Real Valued Random Variables   crrv 24729
19.3.15.4  Preimage set mapping operator   corvc 24744
19.3.15.5  Distribution Functions   orvcelval 24757
19.3.15.6  Cumulative Distribution Functions   orvclteel 24761
19.3.15.7  Probabilities - example   coinfliplem 24767
19.3.15.8  Bertrand's Ballot Problem   ballotlemoex 24774
19.4  Mathbox for Mario Carneiro
19.4.1  Miscellaneous stuff   quartfull 24827
19.4.2  Zeta function   czeta 24828
19.4.3  Gamma function   clgam 24831
19.4.4  Derangements and the Subfactorial   deranglem 24883
19.4.5  The Erdős-Szekeres theorem   erdszelem1 24908
19.4.6  The Kuratowski closure-complement theorem   kur14lem1 24923
19.4.7  Retracts and sections   cretr 24934
19.4.8  Path-connected and simply connected spaces   cpcon 24937
19.4.9  Covering maps   ccvm 24973
19.4.10  Normal numbers   snmlff 25047
19.4.11  Godel-sets of formulas   cgoe 25051
19.4.12  Models of ZF   cgze 25079
19.4.13  Splitting fields   citr 25093
19.4.14  p-adic number fields   czr 25109
19.5  Mathbox for Paul Chapman
19.5.1  Group homomorphism and isomorphism   ghomgrpilem1 25127
19.5.2  Real and complex numbers (cont.)   climuzcnv 25139
19.5.3  Miscellaneous theorems   elfzm12 25143
*19.6  Mathbox for Drahflow
19.7  Mathbox for Scott Fenton
19.7.1  ZFC Axioms in primitive form   axextprim 25181
19.7.2  Untangled classes   untelirr 25188
19.7.3  Extra propositional calculus theorems   3orel1 25195
19.7.4  Misc. Useful Theorems   nepss 25206
19.7.5  Properties of reals and complexes   sqdivzi 25215
19.7.6  Product sequences   prodf 25246
19.7.7  Non-trivial convergence   ntrivcvg 25256
19.7.8  Complex products   cprod 25262
19.7.9  Finite products   fprod 25298
19.7.10  Infinite products   iprodclim 25342
19.7.11  Falling and Rising Factorial   cfallfac 25351
19.7.12  Factorial limits   faclimlem1 25393
19.7.13  Greatest common divisor and divisibility   pdivsq 25399
19.7.14  Properties of relationships   brtp 25403
19.7.15  Properties of functions and mappings   funpsstri 25420
19.7.16  Epsilon induction   setinds 25436
19.7.17  Ordinal numbers   elpotr 25439
19.7.18  Defined equality axioms   axextdfeq 25456
19.7.19  Hypothesis builders   hbntg 25464
19.7.20  The Predecessor Class   cpred 25469
19.7.21  (Trans)finite Recursion Theorems   tfisg 25510
19.7.22  Well-founded induction   tz6.26 25511
19.7.23  Transitive closure under a relationship   ctrpred 25526
19.7.24  Founded Induction   frmin 25548
19.7.25  Ordering Ordinal Sequences   orderseqlem 25558
19.7.26  Well-founded recursion   cwrecs 25561
19.7.27  Transfinite recursion via Well-founded recursion   tfrALTlem 25588
19.7.28  Well-founded zero, successor, and limits   cwsuc 25592
19.7.29  Founded Recursion   frr3g 25612
19.7.30  Surreal Numbers   csur 25626
19.7.31  Surreal Numbers: Ordering   sltsolem1 25654
19.7.32  Surreal Numbers: Birthday Function   bdayfo 25661
19.7.33  Surreal Numbers: Density   fvnobday 25668
19.7.34  Surreal Numbers: Density   nodenselem3 25669
19.7.35  Surreal Numbers: Upper and Lower Bounds   nobndlem1 25678
19.7.36  Surreal Numbers: Full-Eta Property   nofulllem1 25688
19.7.37  Symmetric difference   csymdif 25693
19.7.38  Quantifier-free definitions   ctxp 25705
19.7.39  Alternate ordered pairs   caltop 25832
19.7.40  Tarskian geometry   cee 25858
19.7.41  Tarski's axioms for geometry   axdimuniq 25883
19.7.42  Congruence properties   cofs 25947
19.7.43  Betweenness properties   btwntriv2 25977
19.7.44  Segment Transportation   ctransport 25994
19.7.45  Properties relating betweenness and congruence   cifs 26000
19.7.46  Connectivity of betweenness   btwnconn1lem1 26052
19.7.47  Segment less than or equal to   csegle 26071
19.7.48  Outside of relationship   coutsideof 26084
19.7.49  Lines and Rays   cline2 26099
19.7.50  Bernoulli polynomials and sums of k-th powers   cbp 26123
19.7.51  Rank theorems   rankung 26138
19.7.52  Hereditarily Finite Sets   chf 26144
19.8  Mathbox for Anthony Hart
19.8.1  Propositional Calculus   tb-ax1 26159
19.8.2  Predicate Calculus   quantriv 26181
19.8.3  Misc. Single Axiom Systems   meran1 26192
19.8.4  Connective Symmetry   negsym1 26198
19.9  Mathbox for Chen-Pang He
19.9.1  Ordinal topology   ontopbas 26209
19.10  Mathbox for Jeff Hoffman
19.10.1  Inferences for finite induction on generic function values   fveleq 26232
19.10.2  gdc.mm   nnssi2 26236
*19.11  Mathbox for Wolf Lammen
19.12  Mathbox for Brendan Leahy
19.13  Mathbox for Jeff Hankins
19.13.1  Miscellany   a1i13 26336
19.13.2  Basic topological facts   topbnd 26365
19.13.3  Topology of the real numbers   ivthALT 26376
19.13.4  Refinements   cfne 26377
19.13.5  Neighborhood bases determine topologies   neibastop1 26426
19.13.6  Lattice structure of topologies   topmtcl 26430
19.13.7  Filter bases   fgmin 26437
19.13.8  Directed sets, nets   tailfval 26439
19.14  Mathbox for Jeff Madsen
19.14.1  Logic and set theory   anim12da 26450
19.14.2  Real and complex numbers; integers   filbcmb 26480
19.14.3  Sequences and sums   sdclem2 26484
19.14.4  Topology   subspopn 26496
19.14.5  Metric spaces   metf1o 26499
19.14.6  Continuous maps and homeomorphisms   constcncf 26506
19.14.7  Boundedness   ctotbnd 26513
19.14.8  Isometries   cismty 26545
19.14.9  Heine-Borel Theorem   heibor1lem 26556
19.14.10  Banach Fixed Point Theorem   bfplem1 26569
19.14.11  Euclidean space   crrn 26572
19.14.12  Intervals (continued)   ismrer1 26585
19.14.13  Groups and related structures   exidcl 26589
19.14.14  Rings   rngonegcl 26599
19.14.15  Ring homomorphisms   crnghom 26614
19.14.16  Commutative rings   ccring 26643
19.14.17  Ideals   cidl 26655
19.14.18  Prime rings and integral domains   cprrng 26694
19.14.19  Ideal generators   cigen 26707
19.15  Mathbox for Rodolfo Medina
19.15.1  Partitions   prtlem60 26726
19.16  Mathbox for Stefan O'Rear
19.16.1  Additional elementary logic and set theory   nelss 26770
19.16.2  Additional theory of functions   fninfp 26773
19.16.3  Extensions beyond function theory   gsumvsmul 26783
19.16.4  Additional topology   elrfi 26786
19.16.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26790
19.16.6  Algebraic closure systems   cnacs 26794
19.16.7  Miscellanea 1. Map utilities   constmap 26805
19.16.8  Miscellanea for polynomials   ofmpteq 26814
19.16.9  Multivariate polynomials over the integers   cmzpcl 26816
19.16.10  Miscellanea for Diophantine sets 1   coeq0 26848
19.16.11  Diophantine sets 1: definitions   cdioph 26851
19.16.12  Diophantine sets 2 miscellanea   ellz1 26863
19.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26869
19.16.14  Diophantine sets 3: construction   diophrex 26872
19.16.15  Diophantine sets 4 miscellanea   2sbcrex 26881
19.16.16  Diophantine sets 4: Quantification   rexrabdioph 26892
19.16.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26899
19.16.18  Diophantine sets 6 miscellanea   fz1ssnn 26909
19.16.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26910
19.16.20  Pigeonhole Principle and cardinality helpers   fphpd 26915
19.16.21  A non-closed set of reals is infinite   rencldnfilem 26919
19.16.22  Miscellanea for Lagrange's theorem   icodiamlt 26921
19.16.23  Lagrange's rational approximation theorem   irrapxlem1 26923
19.16.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26930
19.16.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26937
19.16.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26979
19.16.27  Logarithm laws generalized to an arbitrary base   reglogcl 26991
19.16.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26999
19.16.29  X and Y sequences 1: Definition and recurrence laws   crmx 27001
19.16.30  Ordering and induction lemmas for the integers   monotuz 27042
19.16.31  X and Y sequences 2: Order properties   rmxypos 27050
19.16.32  Congruential equations   congtr 27068
19.16.33  Alternating congruential equations   acongid 27078
19.16.34  Additional theorems on integer divisibility   bezoutr 27088
19.16.35  X and Y sequences 3: Divisibility properties   jm2.18 27097
19.16.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 27114
19.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 27124
19.16.38  Uncategorized stuff not associated with a major project   setindtr 27133
19.16.39  More equivalents of the Axiom of Choice   axac10 27142
19.16.40  Finitely generated left modules   clfig 27180
19.16.41  Noetherian left modules I   clnm 27188
19.16.42  Addenda for structure powers   pwssplit0 27202
19.16.43  Direct sum of left modules   cdsmm 27212
19.16.44  Free modules   cfrlm 27227
19.16.45  Every set admits a group structure iff choice   unxpwdom3 27271
19.16.46  Independent sets and families   clindf 27289
19.16.47  Characterization of free modules   lmimlbs 27321
19.16.48  Noetherian rings and left modules II   clnr 27328
19.16.49  Hilbert's Basis Theorem   cldgis 27340
19.16.50  Additional material on polynomials [DEPRECATED]   cmnc 27350
19.16.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27360
19.16.52  Algebraic integers I   citgo 27377
19.16.53  Finite cardinality [SO]   en1uniel 27395
*19.16.54  Words in monoids and ordered group sum   issubmd 27398
19.16.55  Transpositions in the symmetric group   cpmtr 27399
19.16.56  The sign of a permutation   cpsgn 27429
19.16.57  The matrix algebra   cmmul 27454
19.16.58  The determinant   cmdat 27498
19.16.59  Endomorphism algebra   cmend 27504
19.16.60  Subfields   csdrg 27518
19.16.61  Cyclic groups and order   idomrootle 27526
19.16.62  Cyclotomic polynomials   ccytp 27536
19.16.63  Miscellaneous topology   fgraphopab 27544
19.17  Mathbox for Steve Rodriguez
19.17.1  Miscellanea   iso0 27551
19.17.2  Function operations   caofcan 27555
19.17.3  Calculus   lhe4.4ex1a 27561
19.18  Mathbox for Andrew Salmon
19.18.1  Principia Mathematica * 10   pm10.12 27568
19.18.2  Principia Mathematica * 11   2alanimi 27582
19.18.3  Predicate Calculus   sbeqal1 27612
19.18.4  Principia Mathematica * 13 and * 14   pm13.13a 27622
19.18.5  Set Theory   elnev 27653
19.18.6  Arithmetic   addcomgi 27675
19.18.7  Geometry   cplusr 27676
19.19  Mathbox for Glauco Siliprandi
19.19.1  Miscellanea   ssrexf 27698
19.19.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27724
19.19.3  Limits   clim1fr1 27741
19.19.4  Derivatives   dvsinexp 27754
19.19.5  Integrals   ioovolcl 27756
19.19.6  Stone Weierstrass theorem - real version   stoweidlem1 27764
19.19.7  Wallis' product for π   wallispilem1 27828
19.19.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27837
19.20  Mathbox for Saveliy Skresanov
19.20.1  Ceva's theorem   sigarval 27854
19.21  Mathbox for Jarvin Udandy
19.22  Mathbox for Alexander van der Vekens
19.22.1  Double restricted existential uniqueness   r19.32 27959
19.22.1.1  Restricted quantification (extension)   r19.32 27959
19.22.1.2  The empty set (extension)   raaan2 27967
19.22.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 27968
19.22.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 27973
*19.22.2  Alternative definitions of function's and operation's values   wdfat 27985
19.22.2.1  Restricted quantification (extension)   ralbinrald 27991
19.22.2.2  The universal class (extension)   nvelim 27992
19.22.2.3  Introduce the Axiom of Power Sets (extension)   alneu 27993
19.22.2.4  Relations (extension)   sbcrel 27995
19.22.2.5  Functions (extension)   sbcfun 28001
19.22.2.6  Predicate "defined at"   dfateq12d 28007
19.22.2.7  Alternative definition of the value of a function   dfafv2 28010
19.22.2.8  Alternative definition of the value of an operation   aoveq123d 28056
*19.22.3  Auxiliary theorems for graph theory   jaoi3 28086
19.22.3.1  Logical disjunction and conjunction   jaoi3 28086
19.22.3.2  Abbreviated conjunction and disjunction of three wff's   3an4anass 28087
19.22.3.3  Negated equality and membership - extension   eqneqall 28088
19.22.3.4  "Weak deduction theorem" for set theory - extension   ifeqda 28090
19.22.3.5  Power classes - extension   3xpexg 28092
19.22.3.6  Unordered and ordered pairs - extension   nelprd 28093
19.22.3.7  Indexed union and intersection - extension   iunxprg 28107
19.22.3.8  Binary relations - extension   breqn0 28108
19.22.3.9  Ordered-pair class abstractions - extension   elopaelxp 28109
19.22.3.10  Introduce the Axiom of Union - extension   ralxfrd2 28111
19.22.3.11  Relations - extension   resisresindm 28113
19.22.3.12  Functions - extension   sbcfn 28114
19.22.3.13  Operations - extension   oprabv 28127
19.22.3.14  Equinumerosity - extension   resfnfinfin 28133
19.22.3.15  Subtraction - extension   cnm1cn 28135
19.22.3.16  Multiplication - extension   kcnktkm1cn 28136
19.22.3.17  Ordering on reals (cont.) - extension   leaddsuble 28138
19.22.3.18  Nonnegative integers (as a subset of complex numbers) - extension   0mnnnnn0 28142
19.22.3.19  Upper partititions of integers - extension   1eluzge0 28147
19.22.3.20  Finite intervals of integers - extension   ssfz12 28151
19.22.3.21  Half-open integer ranges - extension   elfzonn0 28169
19.22.3.22  The floor (greatest integer) function - extension   nn0nndivcl 28188
19.22.3.23  The modulo (remainder) operation - extension   modvalr 28196
19.22.3.24  The ` # ` (finite set size) function - extension   hashimarn 28210
19.22.3.25  Words over a set - extension   wrdlen1 28223
19.22.3.26  Words over a set - extension (concatenations)   elfzelfzccat 28233
19.22.3.27  Words over a set - extension (subwords)   swrdltnd 28239
19.22.3.28  Words over a set - extension (subwords of subwords)   swrd0swrd 28255
19.22.3.29  Words over a set - extension (subwords of concatenations)   swrdccat3a0 28261
19.22.3.30  Prime numbers: elementary properties - extension   prmgt1 28281
*19.22.3.31  Words over a set - extension (cyclic shift)   ccsh 28288
19.22.4  Graph theory   uhgraedgrnv 28350
19.22.4.1  Undirected hypergraphs   uhgraedgrnv 28350
19.22.4.2  Undirected simple graphs   usisuhgra 28351
19.22.4.3  Neighbors, complete graphs and universal vertices   nbfiusgrafi 28352
19.22.4.4  Walks, Paths and Cycles   wlkn0 28356
19.22.4.5  Walks as words   cwwlk 28383
19.22.4.6  Walks/paths of length 2 as ordered triples   c2wlkot 28410
19.22.4.7  Vertex Degree   usgfidegfi 28449
19.22.4.8  Regular graphs   crgra 28461
*19.22.4.9  Friendship graphs   cfrgra 28476
*19.23  Mathbox for David A. Wheeler
19.23.1  Natural deduction   19.8ad 28558
*19.23.2  Greater than, greater than or equal to.   cge-real 28561
*19.23.3  Hyperbolic trig functions   csinh 28571
*19.23.4  Reciprocal trig functions (sec, csc, cot)   csec 28582
*19.23.5  Identities for "if"   ifnmfalse 28604
19.23.6  Not-member-of   AnelBC 28605
*19.23.7  Decimal point   cdp2 28606
19.23.8  Signum (sgn or sign) function   csgn 28614
19.23.9  Ceiling function   ccei 28624
19.23.10  Logarithms generalized to arbitrary base using ` logb `   ene0 28628
*19.23.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 28631
*19.23.12  Miscellaneous   5m4e1 28633
19.24  Mathbox for Alan Sare
19.24.2  Supplementary unification deductions   biimp 28662
19.24.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 28681
19.24.4  What is Virtual Deduction?   wvd1 28758
19.24.5  Virtual Deduction Theorems   df-vd1 28759
19.24.6  Theorems proved using virtual deduction   trsspwALT 29029
19.24.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 29056
19.24.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 29123
19.24.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 29127
19.24.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 29134
*19.24.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 29137
*19.25  Mathbox for Jonathan Ben-Naim
19.25.1  First order logic and set theory   bnj170 29160
19.25.2  Well founded induction and recursion   bnj110 29327
19.25.3  The existence of a minimal element in certain classes   bnj69 29477
19.25.4  Well-founded induction   bnj1204 29479
19.25.5  Well-founded recursion, part 1 of 3   bnj60 29529
19.25.6  Well-founded recursion, part 2 of 3   bnj1500 29535
19.25.7  Well-founded recursion, part 3 of 3   bnj1522 29539
*19.26  Mathbox for Norm Megill
*19.26.1  Experiments to study ax-7 unbundling   ax-7v 29540
19.26.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)   ax-7v 29540
19.26.1.2  Theorems derived from ax-7 (suffix OLD7)   ax-7OLD7 29776
19.26.2  Miscellanea   cnaddcom 29867
19.26.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 29870
19.26.4  Functionals and kernels of a left vector space (or module)   clfn 29953
19.26.5  Opposite rings and dual vector spaces   cld 30019
19.26.6  Ortholattices and orthomodular lattices   cops 30068
19.26.7  Atomic lattices with covering property   ccvr 30158
19.26.8  Hilbert lattices   chlt 30246
19.26.9  Projective geometries based on Hilbert lattices   clln 30386
19.26.10  Construction of a vector space from a Hilbert lattice   cdlema1N 30686
19.26.11  Construction of involution and inner product from a Hilbert lattice   clpoN 32376

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