Home Metamath Proof ExplorerTheorem List (Table of Contents) < Wrap  Next > Browser slow? Try the Unicode version.

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

PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
1.1  Pre-logic
1.1.1  Inferences for assisting proof development   dummylink 1
1.2  Propositional calculus
1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
1.2.2  The axioms of propositional calculus   ax-1 5
1.2.3  Logical implication   mp2 9
1.2.4  Logical negation   con4d 99
1.2.5  Logical equivalence   wb 177
1.2.6  Logical disjunction and conjunction   wo 358
1.2.7  Miscellaneous theorems of propositional calculus   pm5.21nd 869
1.2.8  Abbreviated conjunction and disjunction of three wff's   w3o 935
1.2.9  Logical 'nand' (Sheffer stroke)   wnan 1293
1.2.10  Logical 'xor'   wxo 1310
1.2.11  True and false constants   wtru 1322
1.2.12  Truth tables   truantru 1342
1.2.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1368
1.2.14  Half-adders and full adders in propositional calculus   whad 1384
1.3  Other axiomatizations of classical propositional calculus
1.3.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1410
1.3.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1429
1.3.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1440
1.3.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1446
1.3.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1465
1.3.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1469
1.3.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1484
1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1507
1.3.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1520
1.3.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1539
1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
1.4.1  Universal quantifier; define "exists" and "not free"   wal 1546
1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1552
1.4.3  Axiom scheme ax-5 (Quantified Implication)   ax-5 1563
1.4.4  Axiom scheme ax-17 (Distinctness) - first use of \$d   ax-17 1623
1.4.5  Equality predicate; define substitution   cv 1648
1.4.6  Axiom scheme ax-9 (Existence)   ax-9 1662
1.4.7  Axiom scheme ax-8 (Equality)   ax-8 1683
1.4.8  Membership predicate   wcel 1721
1.4.9  Axiom scheme ax-13 (Left Equality for Binary Predicate)   ax-13 1723
1.4.10  Axiom scheme ax-14 (Right Equality for Binary Predicate)   ax-14 1725
1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1727
1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
1.5.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1740
1.5.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1745
1.5.3  Axiom scheme ax-11 (Substitution)   ax-11 1757
1.5.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1946
1.6  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
1.6.1  Obsolete schemes ax-5o ax-4 ax-6o ax-9o ax-10o ax-10 ax-11o ax-12o ax-15 ax-16   ax-4 2185
1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o   ax4 2195
1.6.3  Legacy theorems using obsolete axioms   ax17o 2207
1.7  Existential uniqueness
1.8  Other axiomatizations related to classical predicate calculus
1.8.1  Predicate calculus with all distinct variables   ax-7d 2345
1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2351
1.8.3  Intuitionistic logic   axi4 2375
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 2385
2.1.2  Class abstractions (a.k.a. class builders)   cab 2390
2.1.3  Class form not-free predicate   wnfc 2527
2.1.4  Negated equality and membership   wne 2567
2.1.4.1  Negated equality   nne 2571
2.1.4.2  Negated membership   neleq1 2660
2.1.5  Restricted quantification   wral 2666
2.1.6  The universal class   cvv 2916
2.1.7  Conditional equality (experimental)   wcdeq 3104
2.1.8  Russell's Paradox   ru 3120
2.1.9  Proper substitution of classes for sets   wsbc 3121
2.1.10  Proper substitution of classes for sets into classes   csb 3211
2.1.11  Define basic set operations and relations   cdif 3277
2.1.12  Subclasses and subsets   df-ss 3294
2.1.13  The difference, union, and intersection of two classes   difeq1 3418
2.1.13.1  The difference of two classes   difeq1 3418
2.1.13.2  The union of two classes   elun 3448
2.1.13.3  The intersection of two classes   elin 3490
2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3531
2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdif2 3567
2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3581
2.1.14  The empty set   c0 3588
2.1.15  "Weak deduction theorem" for set theory   cif 3699
2.1.16  Power classes   cpw 3759
2.1.17  Unordered and ordered pairs   csn 3774
2.1.18  The union of a class   cuni 3975
2.1.19  The intersection of a class   cint 4010
2.1.20  Indexed union and intersection   ciun 4053
2.1.21  Disjointness   wdisj 4142
2.1.22  Binary relations   wbr 4172
2.1.23  Ordered-pair class abstractions (class builders)   copab 4225
2.1.24  Transitive classes   wtr 4262
2.2  ZF Set Theory - add the Axiom of Replacement
2.2.1  Introduce the Axiom of Replacement   ax-rep 4280
2.2.2  Derive the Axiom of Separation   axsep 4289
2.2.3  Derive the Null Set Axiom   zfnuleu 4295
2.2.4  Theorems requiring subset and intersection existence   nalset 4300
2.2.5  Theorems requiring empty set existence   class2set 4327
2.3  ZF Set Theory - add the Axiom of Power Sets
2.3.1  Introduce the Axiom of Power Sets   ax-pow 4337
2.3.2  Derive the Axiom of Pairing   zfpair 4361
2.3.3  Ordered pair theorem   opnz 4392
2.3.4  Ordered-pair class abstractions (cont.)   opabid 4421
2.3.5  Power class of union and intersection   pwin 4447
2.3.6  Epsilon and identity relations   cep 4452
2.3.7  Partial and complete ordering   wpo 4461
2.3.8  Founded and well-ordering relations   wfr 4498
2.3.9  Ordinals   word 4540
2.4  ZF Set Theory - add the Axiom of Union
2.4.1  Introduce the Axiom of Union   ax-un 4660
2.4.2  Ordinals (continued)   ordon 4722
2.4.3  Transfinite induction   tfi 4792
2.4.4  The natural numbers (i.e. finite ordinals)   com 4804
2.4.5  Peano's postulates   peano1 4823
2.4.6  Finite induction (for finite ordinals)   find 4829
2.4.7  Relations   cxp 4835
2.4.8  Definite description binder (inverted iota)   cio 5375
2.4.9  Functions   wfun 5407
2.4.10  Operations   co 6040
2.4.11  "Maps to" notation   elmpt2cl 6247
2.4.12  Function operation   cof 6262
2.4.13  First and second members of an ordered pair   c1st 6306
2.4.14  Special "Maps to" operations   mpt2xopn0yelv 6423
2.4.15  Function transposition   ctpos 6437
2.4.16  Curry and uncurry   ccur 6476
2.4.17  Proper subset relation   crpss 6480
2.4.18  Iota properties   fvopab5 6493
2.4.19  Cantor's Theorem   canth 6498
2.4.20  Undefined values and restricted iota (description binder)   cund 6500
2.4.21  Functions on ordinals; strictly monotone ordinal functions   iunon 6559
2.4.22  "Strong" transfinite recursion   crecs 6591
2.4.23  Recursive definition generator   crdg 6626
2.4.24  Finite recursion   frfnom 6651
2.4.25  Abian's "most fundamental" fixed point theorem   abianfplem 6674
2.4.26  Ordinal arithmetic   c1o 6676
2.4.27  Natural number arithmetic   nna0 6806
2.4.28  Equivalence relations and classes   wer 6861
2.4.29  The mapping operation   cmap 6977
2.4.30  Infinite Cartesian products   cixp 7022
2.4.31  Equinumerosity   cen 7065
2.4.32  Schroeder-Bernstein Theorem   sbthlem1 7176
2.4.33  Equinumerosity (cont.)   xpf1o 7228
2.4.34  Pigeonhole Principle   phplem1 7245
2.4.35  Finite sets   onomeneq 7255
2.4.36  Finite intersections   cfi 7373
2.4.37  Hall's marriage theorem   marypha1lem 7396
2.4.38  Supremum   csup 7403
2.4.39  Ordinal isomorphism, Hartog's theorem   coi 7434
2.4.40  Hartogs function, order types, weak dominance   char 7480
2.5  ZF Set Theory - add the Axiom of Regularity
2.5.1  Introduce the Axiom of Regularity   ax-reg 7516
2.5.2  Axiom of Infinity equivalents   inf0 7532
2.6  ZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity   ax-inf 7549
2.6.2  Existence of omega (the set of natural numbers)   omex 7554
2.6.3  Cantor normal form   ccnf 7572
2.6.4  Transitive closure   trcl 7620
2.6.5  Rank   cr1 7644
2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7765
2.6.7  Cardinal numbers   ccrd 7778
2.6.8  Axiom of Choice equivalents   wac 7952
2.6.9  Cardinal number arithmetic   ccda 8003
2.6.10  The Ackermann bijection   ackbij2lem1 8055
2.6.11  Cofinality (without Axiom of Choice)   cflem 8082
2.6.12  Eight inequivalent definitions of finite set   sornom 8113
2.6.13  Hereditarily size-limited sets without Choice   itunifval 8252
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 8295
3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8330
3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8377
3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8405
3.2.5  Cofinality using Axiom of Choice   alephreg 8413
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 8513
4.1.2  Weak universes   cwun 8531
4.1.3  Tarski's classes   ctsk 8579
4.1.4  Grothendieck's universes   cgru 8621
4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8654
4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8657
4.2.3  Tarski map function   ctskm 8668
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 8675
5.1.2  Final derivation of real and complex number postulates   axaddf 8976
5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9002
5.2  Derive the basic properties from the field axioms
5.2.1  Some deductions from the field axioms for complex numbers   cnex 9027
5.2.2  Infinity and the extended real number system   cpnf 9073
5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9103
5.2.4  Ordering on reals   lttr 9108
5.2.5  Initial properties of the complex numbers   mul12 9188
5.3  Real and complex numbers - basic operations
5.3.2  Subtraction   cmin 9247
5.3.3  Multiplication   muladd 9422
5.3.4  Ordering on reals (cont.)   gt0ne0 9449
5.3.5  Reciprocals   ixi 9607
5.3.6  Division   cdiv 9633
5.3.7  Ordering on reals (cont.)   elimgt0 9802
5.3.8  Completeness Axiom and Suprema   fimaxre 9911
5.3.9  Imaginary and complex number properties   inelr 9946
5.3.10  Function operation analogue theorems   ofsubeq0 9953
5.4  Integer sets
5.4.1  Natural numbers (as a subset of complex numbers)   cn 9956
5.4.2  Principle of mathematical induction   nnind 9974
5.4.3  Decimal representation of numbers   c2 10005
5.4.4  Some properties of specific numbers   0p1e1 10049
5.4.5  The Archimedean property   nnunb 10173
5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 10177
5.4.7  Integers (as a subset of complex numbers)   cz 10238
5.4.8  Decimal arithmetic   cdc 10338
5.4.9  Upper partititions of integers   cuz 10444
5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10525
5.4.11  Rational numbers (as a subset of complex numbers)   cq 10530
5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10556
5.5  Order sets
5.5.1  Positive reals (as a subset of complex numbers)   crp 10568
5.5.2  Infinity and the extended real number system (cont.)   cxne 10663
5.5.3  Supremum on the extended reals   xrsupexmnf 10839
5.5.4  Real number intervals   cioo 10872
5.5.5  Finite intervals of integers   cfz 10999
5.5.6  Half-open integer ranges   cfzo 11090
5.6  Elementary integer functions
5.6.1  The floor (greatest integer) function   cfl 11156
5.6.2  The modulo (remainder) operation   cmo 11205
5.6.3  The infinite sequence builder "seq"   om2uz0i 11242
5.6.4  Integer powers   cexp 11337
5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11515
5.6.6  Factorial function   cfa 11521
5.6.7  The binomial coefficient operation   cbc 11548
5.6.8  The ` # ` (finite set size) function   chash 11573
5.6.8.1  Finite induction on the size of the first component of a binary relation   brfi1indlem 11669
5.6.9  Words over a set   cword 11672
5.6.10  Longer string literals   cs2 11760
5.7  Elementary real and complex functions
5.7.1  The "shift" operation   cshi 11836
5.7.2  Real and imaginary parts; conjugate   ccj 11856
5.7.3  Square root; absolute value   csqr 11993
5.8  Elementary limits and convergence
5.8.1  Superior limit (lim sup)   clsp 12219
5.8.2  Limits   cli 12233
5.8.3  Finite and infinite sums   csu 12434
5.8.4  The binomial theorem   binomlem 12563
5.8.5  The inclusion/exclusion principle   incexclem 12571
5.8.6  Infinite sums (cont.)   isumshft 12574
5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12587
5.8.8  Arithmetic series   arisum 12594
5.8.9  Geometric series   expcnv 12598
5.8.10  Ratio test for infinite series convergence   cvgrat 12615
5.8.11  Mertens' theorem   mertenslem1 12616
5.9  Elementary trigonometry
5.9.1  The exponential, sine, and cosine functions   ce 12619
5.9.2  _e is irrational   eirrlem 12758
5.10  Cardinality of real and complex number subsets
5.10.1  Countability of integers and rationals   xpnnen 12763
5.10.2  The reals are uncountable   rpnnen2lem1 12769
PART 6  ELEMENTARY NUMBER THEORY
6.1  Elementary properties of divisibility
6.1.1  Irrationality of square root of 2   sqr2irrlem 12802
6.1.2  Some Number sets are chains of proper subsets   nthruc 12805
6.1.3  The divides relation   cdivides 12807
6.1.4  The division algorithm   divalglem0 12868
6.1.5  Bit sequences   cbits 12886
6.1.6  The greatest common divisor operator   cgcd 12961
6.1.7  Bézout's identity   bezoutlem1 12993
6.1.8  Algorithms   nn0seqcvgd 13016
6.1.9  Euclid's Algorithm   eucalgval2 13027
6.2  Elementary prime number theory
6.2.1  Elementary properties   cprime 13034
6.2.2  Properties of the canonical representation of a rational   cnumer 13080
6.2.3  Euler's theorem   codz 13107
6.2.4  Pythagorean Triples   coprimeprodsq 13138
6.2.5  The prime count function   cpc 13165
6.2.6  Pocklington's theorem   prmpwdvds 13227
6.2.7  Infinite primes theorem   unbenlem 13231
6.2.8  Sum of prime reciprocals   prmreclem1 13239
6.2.9  Fundamental theorem of arithmetic   1arithlem1 13246
6.2.10  Lagrange's four-square theorem   cgz 13252
6.2.11  Van der Waerden's theorem   cvdwa 13288
6.2.12  Ramsey's theorem   cram 13322
6.2.13  Decimal arithmetic (cont.)   dec2dvds 13354
6.2.14  Specific prime numbers   4nprm 13382
6.2.15  Very large primes   1259lem1 13405
PART 7  BASIC STRUCTURES
7.1  Extensible structures
7.1.1  Basic definitions   cstr 13420
7.1.2  Slot definitions   cplusg 13484
7.1.3  Definition of the structure product   crest 13603
7.1.4  Definition of the structure quotient   cordt 13676
7.2  Moore spaces
7.2.1  Moore closures   mrcflem 13786
7.2.2  Independent sets in a Moore system   mrisval 13810
7.2.3  Algebraic closure systems   isacs 13831
PART 8  BASIC CATEGORY THEORY
8.1  Categories
8.1.1  Categories   ccat 13844
8.1.2  Opposite category   coppc 13892
8.1.3  Monomorphisms and epimorphisms   cmon 13909
8.1.4  Sections, inverses, isomorphisms   csect 13925
8.1.5  Subcategories   cssc 13962
8.1.6  Functors   cfunc 14006
8.1.7  Full & faithful functors   cful 14054
8.1.8  Natural transformations and the functor category   cnat 14093
8.2  Arrows (disjointified hom-sets)
8.2.1  Identity and composition for arrows   cida 14163
8.3  Examples of categories
8.3.1  The category of sets   csetc 14185
8.3.2  The category of categories   ccatc 14204
8.4  Categorical constructions
8.4.1  Product of categories   cxpc 14220
8.4.2  Functor evaluation   cevlf 14261
8.4.3  Hom functor   chof 14300
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 14352
9.2.2  Lattices   clat 14429
9.2.3  The dual of an ordered set   codu 14510
9.2.4  Subset order structures   cipo 14532
9.2.5  Distributive lattices   latmass 14569
9.2.6  Posets and lattices as relations   cps 14579
9.2.7  Directed sets, nets   cdir 14628
PART 10  BASIC ALGEBRAIC STRUCTURES
10.1  Monoids
10.1.1  Definition and basic properties   cmnd 14639
10.1.2  Monoid homomorphisms and submonoids   cmhm 14691
10.1.3  Ordered group sum operation   gsumvallem1 14726
10.1.4  Free monoids   cfrmd 14747
10.2  Groups
10.2.1  Definition and basic properties   df-grp 14767
10.2.2  Subgroups and Quotient groups   csubg 14893
10.2.3  Elementary theory of group homomorphisms   cghm 14958
10.2.4  Isomorphisms of groups   cgim 14999
10.2.5  Group actions   cga 15021
10.2.6  Symmetry groups and Cayley's Theorem   csymg 15047
10.2.7  Centralizers and centers   ccntz 15069
10.2.8  The opposite group   coppg 15096
10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 15118
10.2.10  Direct products   clsm 15223
10.2.11  Free groups   cefg 15293
10.3  Abelian groups
10.3.1  Definition and basic properties   ccmn 15367
10.3.2  Cyclic groups   ccyg 15442
10.3.3  Group sum operation   gsumval3a 15467
10.3.4  Internal direct products   cdprd 15509
10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15578
10.4  Rings
10.4.1  Multiplicative Group   cmgp 15603
10.4.2  Definition and basic properties   crg 15615
10.4.3  Opposite ring   coppr 15682
10.4.4  Divisibility   cdsr 15698
10.4.5  Ring homomorphisms   crh 15772
10.5  Division rings and fields
10.5.1  Definition and basic properties   cdr 15790
10.5.2  Subrings of a ring   csubrg 15819
10.5.3  Absolute value (abstract algebra)   cabv 15859
10.5.4  Star rings   cstf 15886
10.6  Left modules
10.6.1  Definition and basic properties   clmod 15905
10.6.2  Subspaces and spans in a left module   clss 15963
10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 16050
10.6.4  Subspace sum; bases for a left module   clbs 16101
10.7  Vector spaces
10.7.1  Definition and basic properties   clvec 16129
10.8  Ideals
10.8.1  The subring algebra; ideals   csra 16195
10.8.2  Two-sided ideals and quotient rings   c2idl 16257
10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 16267
10.8.4  Nonzero rings   cnzr 16283
10.8.5  Left regular elements. More kinds of rings   crlreg 16294
10.9  Associative algebras
10.9.1  Definition and basic properties   casa 16324
10.10  Abstract multivariate polynomials
10.10.1  Definition and basic properties   cmps 16361
10.10.2  Polynomial evaluation   evlslem4 16519
10.10.3  Univariate polynomials   cps1 16524
10.11  The complex numbers as an extensible structure
10.11.1  Definition and basic properties   cpsmet 16640
10.11.2  Algebraic constructions based on the complexes   czrh 16733
10.12  Hilbert spaces
10.12.1  Definition and basic properties   cphl 16810
10.12.2  Orthocomplements and closed subspaces   cocv 16842
10.12.3  Orthogonal projection and orthonormal bases   cpj 16882
PART 11  BASIC TOPOLOGY
11.1  Topology
11.1.1  Topological spaces   ctop 16913
11.1.2  TopBases for topologies   isbasisg 16967
11.1.3  Examples of topologies   distop 17015
11.1.4  Closure and interior   ccld 17035
11.1.5  Neighborhoods   cnei 17116
11.1.6  Limit points and perfect sets   clp 17153
11.1.7  Subspace topologies   restrcl 17175
11.1.8  Order topology   ordtbaslem 17206
11.1.9  Limits and continuity in topological spaces   ccn 17242
11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17324
11.1.11  Compactness   ccmp 17403
11.1.12  Connectedness   ccon 17427
11.1.13  First- and second-countability   c1stc 17453
11.1.14  Local topological properties   clly 17480
11.1.15  Compactly generated spaces   ckgen 17518
11.1.16  Product topologies   ctx 17545
11.1.17  Continuous function-builders   cnmptid 17646
11.1.18  Quotient maps and quotient topology   ckq 17678
11.1.19  Homeomorphisms   chmeo 17738
11.2  Filters and filter bases
11.2.1  Filter bases   elmptrab 17812
11.2.2  Filters   cfil 17830
11.2.3  Ultrafilters   cufil 17884
11.2.4  Filter limits   cfm 17918
11.2.5  Extension by continuity   ccnext 18043
11.2.6  Topological groups   ctmd 18053
11.2.7  Infinite group sum on topological groups   ctsu 18108
11.2.8  Topological rings, fields, vector spaces   ctrg 18138
11.3  Uniform Stuctures and Spaces
11.3.1  Uniform structures   cust 18182
11.3.2  The topology induced by an uniform structure   cutop 18213
11.3.3  Uniform Spaces   cuss 18236
11.3.4  Uniform continuity   cucn 18258
11.3.5  Cauchy filters in uniform spaces   ccfilu 18269
11.3.6  Complete uniform spaces   ccusp 18280
11.4  Metric spaces
11.4.1  Pseudometric spaces   ispsmet 18288
11.4.2  Basic metric space properties   cxme 18300
11.4.3  Metric space balls   blfvalps 18366
11.4.4  Open sets of a metric space   mopnval 18421
11.4.5  Continuity in metric spaces   metcnp3 18523
11.4.6  The uniform structure generated by a metric   metuvalOLD 18532
11.4.7  Examples of metric spaces   dscmet 18573
11.4.8  Normed algebraic structures   cnm 18577
11.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 18692
11.4.10  Topology on the reals   qtopbaslem 18745
11.4.11  Topological definitions using the reals   cii 18858
11.4.12  Path homotopy   chtpy 18945
11.4.13  The fundamental group   cpco 18978
11.5  Complex metric vector spaces
11.5.1  Complex left modules   cclm 19040
11.5.2  Complex pre-Hilbert space   ccph 19082
11.5.3  Convergence and completeness   ccfil 19158
11.5.4  Baire's Category Theorem   bcthlem1 19230
11.5.5  Banach spaces and complex Hilbert spaces   ccms 19238
11.5.6  Minimizing Vector Theorem   minveclem1 19278
11.5.7  Projection Theorem   pjthlem1 19291
PART 12  BASIC REAL AND COMPLEX ANALYSIS
12.1  Continuity
12.1.1  Intermediate value theorem   pmltpclem1 19298
12.2  Integrals
12.2.1  Lebesgue measure   covol 19312
12.2.2  Lebesgue integration   cmbf 19459
12.3  Derivatives
12.3.1  Real and complex differentiation   climc 19702
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
13.1  Polynomials
13.1.1  Abstract polynomials, continued   evlslem6 19887
13.1.2  Polynomial degrees   cmdg 19929
13.1.3  The division algorithm for univariate polynomials   cmn1 20001
13.1.4  Elementary properties of complex polynomials   cply 20056
13.1.5  The division algorithm for polynomials   cquot 20160
13.1.6  Algebraic numbers   caa 20184
13.1.7  Liouville's approximation theorem   aalioulem1 20202
13.2  Sequences and series
13.2.1  Taylor polynomials and Taylor's theorem   ctayl 20222
13.2.2  Uniform convergence   culm 20245
13.2.3  Power series   pserval 20279
13.3  Basic trigonometry
13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 20312
13.3.2  Properties of pi = 3.14159...   pilem1 20320
13.3.3  Mapping of the exponential function   efgh 20396
13.3.4  The natural logarithm on complex numbers   clog 20405
13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20596
13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20632
13.3.7  Inverse trigonometric functions   casin 20655
13.3.8  The Birthday Problem   log2ublem1 20739
13.3.9  Areas in R^2   carea 20747
13.3.10  More miscellaneous converging sequences   rlimcnp 20757
13.3.11  Inequality of arithmetic and geometric means   cvxcl 20776
13.3.12  Euler-Mascheroni constant   cem 20783
13.4  Basic number theory
13.4.1  Wilson's theorem   wilthlem1 20804
13.4.2  The Fundamental Theorem of Algebra   ftalem1 20808
13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20816
13.4.4  Number-theoretical functions   ccht 20826
13.4.5  Perfect Number Theorem   mersenne 20964
13.4.6  Characters of Z/nZ   cdchr 20969
13.4.7  Bertrand's postulate   bcctr 21012
13.4.8  Legendre symbol   clgs 21031
13.4.9  Quadratic reciprocity   lgseisenlem1 21086
13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 21100
13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 21116
13.4.12  The Prime Number Theorem   mudivsum 21177
13.4.13  Ostrowski's theorem   abvcxp 21262
PART 14  GRAPH THEORY
14.1  Undirected graphs - basics
14.1.1  Undirected hypergraphs   cuhg 21287
14.1.2  Undirected multigraphs   cumg 21300
14.1.3  Undirected simple graphs   cuslg 21317
14.1.3.1  Undirected simple graphs - basics   cuslg 21317
14.1.3.2  Undirected simple graphs - examples   usgraexvlem 21367
14.1.3.3  Finite undirected simple graphs   fiusgraedgfi 21374
14.1.4  Neighbors, complete graphs and universal vertices   cnbgra 21383
14.1.4.1  Neighbors   nbgraop 21389
14.1.4.2  Complete graphs   iscusgra 21418
14.1.4.3  Universal vertices   isuvtx 21450
14.1.5  Walks, paths and cycles   cwalk 21459
14.1.5.1  Walks and trails   wlks 21479
14.1.5.2  Paths and simple paths   pths 21519
14.1.5.3  Circuits and cycles   crcts 21562
14.1.5.4  Connected graphs   cconngra 21609
14.1.6  Vertex Degree   cvdg 21617
14.2  Eulerian paths and the Konigsberg Bridge problem
14.2.1  Eulerian paths   ceup 21637
14.2.2  The Konigsberg Bridge problem   vdeg0i 21657
PART 15  GUIDES AND MISCELLANEA
15.1  Guides (conventions, explanations, and examples)
15.1.1  Conventions   conventions 21663
15.1.2  Natural deduction   natded 21664
15.1.3  Natural deduction examples   ex-natded5.2 21665
15.1.4  Definitional examples   ex-or 21682
15.2  Humor
15.2.1  April Fool's theorem   avril1 21710
15.3  (Future - to be reviewed and classified)
15.3.1  Planar incidence geometry   cplig 21716
15.3.2  Algebra preliminaries   crpm 21721
15.3.3  Transitive closure   ctcl 21723
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 21727
16.1.2  Definition and basic properties of Abelian groups   cablo 21822
16.1.3  Subgroups   csubgo 21842
16.1.4  Operation properties   cass 21853
16.1.5  Group-like structures   cmagm 21859
16.1.6  Examples of Abelian groups   ablosn 21888
16.1.7  Group homomorphism and isomorphism   cghom 21898
16.2  Additional material on rings and fields
16.2.1  Definition and basic properties   crngo 21916
16.2.2  Examples of rings   cnrngo 21944
16.2.3  Division Rings   cdrng 21946
16.2.4  Star Fields   csfld 21949
16.2.5  Fields and Rings   ccm2 21951
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
17.1  Complex vector spaces
17.1.1  Definition and basic properties   cvc 21977
17.1.2  Examples of complex vector spaces   cncvc 22015
17.2  Normed complex vector spaces
17.2.1  Definition and basic properties   cnv 22016
17.2.2  Examples of normed complex vector spaces   cnnv 22121
17.2.3  Induced metric of a normed complex vector space   imsval 22130
17.2.4  Inner product   cdip 22149
17.2.5  Subspaces   css 22173
17.3  Operators on complex vector spaces
17.3.1  Definitions and basic properties   clno 22194
17.4  Inner product (pre-Hilbert) spaces
17.4.1  Definition and basic properties   ccphlo 22266
17.4.2  Examples of pre-Hilbert spaces   cncph 22273
17.4.3  Properties of pre-Hilbert spaces   isph 22276
17.5  Complex Banach spaces
17.5.1  Definition and basic properties   ccbn 22317
17.5.2  Examples of complex Banach spaces   cnbn 22324
17.5.3  Uniform Boundedness Theorem   ubthlem1 22325
17.5.4  Minimizing Vector Theorem   minvecolem1 22329
17.6  Complex Hilbert spaces
17.6.1  Definition and basic properties   chlo 22340
17.6.2  Standard axioms for a complex Hilbert space   hlex 22353
17.6.3  Examples of complex Hilbert spaces   cnchl 22371
17.6.4  Subspaces   ssphl 22372
17.6.5  Hellinger-Toeplitz Theorem   htthlem 22373
PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
18.1  Axiomatization of complex pre-Hilbert spaces
18.1.1  Basic Hilbert space definitions   chil 22375
18.1.2  Preliminary ZFC lemmas   df-hnorm 22424
18.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 22437
18.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 22455
18.1.5  Vector operations   hvmulex 22467
18.1.6  Inner product postulates for a Hilbert space   ax-hfi 22534
18.2  Inner product and norms
18.2.1  Inner product   his5 22541
18.2.2  Norms   dfhnorm2 22577
18.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 22615
18.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 22634
18.3  Cauchy sequences and completeness axiom
18.3.1  Cauchy sequences and limits   hcau 22639
18.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 22649
18.3.3  Completeness postulate for a Hilbert space   ax-hcompl 22657
18.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 22658
18.4  Subspaces and projections
18.4.1  Subspaces   df-sh 22662
18.4.2  Closed subspaces   df-ch 22677
18.4.3  Orthocomplements   df-oc 22707
18.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 22763
18.4.5  Projection theorem   pjhthlem1 22846
18.4.6  Projectors   df-pjh 22850
18.5  Properties of Hilbert subspaces
18.5.1  Orthomodular law   omlsilem 22857
18.5.2  Projectors (cont.)   pjhtheu2 22871
18.5.3  Hilbert lattice operations   sh0le 22895
18.5.4  Span (cont.) and one-dimensional subspaces   spansn0 22996
18.5.5  Commutes relation for Hilbert lattice elements   df-cm 23038
18.5.6  Foulis-Holland theorem   fh1 23073
18.5.7  Quantum Logic Explorer axioms   qlax1i 23082
18.5.8  Orthogonal subspaces   chscllem1 23092
18.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 23109
18.5.10  Projectors (cont.)   pjorthi 23124
18.5.11  Mayet's equation E_3   mayete3i 23183
18.6  Operators on Hilbert spaces
18.6.1  Operator sum, difference, and scalar multiplication   df-hosum 23186
18.6.2  Zero and identity operators   df-h0op 23204
18.6.3  Operations on Hilbert space operators   hoaddcl 23214
18.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 23295
18.6.5  Linear and continuous functionals and norms   df-nmfn 23301
18.6.7  Dirac bra-ket notation   df-bra 23306
18.6.8  Positive operators   df-leop 23308
18.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 23309
18.6.10  Theorems about operators and functionals   nmopval 23312
18.6.11  Riesz lemma   riesz3i 23518
18.6.13  Quantum computation error bound theorem   unierri 23560
18.6.14  Dirac bra-ket notation (cont.)   branmfn 23561
18.6.15  Positive operators (cont.)   leopg 23578
18.6.16  Projectors as operators   pjhmopi 23602
18.7  States on a Hilbert lattice and Godowski's equation
18.7.1  States on a Hilbert lattice   df-st 23667
18.7.2  Godowski's equation   golem1 23727
18.8  Cover relation, atoms, exchange axiom, and modular symmetry
18.8.1  Covers relation; modular pairs   df-cv 23735
18.8.2  Atoms   df-at 23794
18.8.3  Superposition principle   superpos 23810
18.8.4  Atoms, exchange and covering properties, atomicity   chcv1 23811
18.8.5  Irreducibility   chirredlem1 23846
18.8.6  Atoms (cont.)   atcvat3i 23852
18.8.7  Modular symmetry   mdsymlem1 23859
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
19.1  Mathboxes for user contributions
19.1.1  Mathbox guidelines   mathbox 23898
19.2  Mathbox for Stefan Allan
19.3  Mathbox for Thierry Arnoux
19.3.1  Propositional Calculus - misc additions   bian1d 23903
19.3.2  Predicate Calculus   abeq2f 23913
19.3.2.1  Predicate Calculus - misc additions   abeq2f 23913
19.3.2.2  Restricted quantification - misc additions   reximddv 23915
19.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 23924
19.3.2.4  Existential "at most one" - misc additions   mo5f 23925
19.3.2.5  Existential uniqueness - misc additions   2reuswap2 23928
19.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 23932
19.3.3  General Set Theory   ceqsexv2d 23938
19.3.3.1  Class abstractions (a.k.a. class builders)   ceqsexv2d 23938
19.3.3.2  Image Sets   abrexdomjm 23941
19.3.3.3  Set relations and operations - misc additions   eqri 23947
19.3.3.4  Unordered pairs   elpreq 23952
19.3.3.5  Conditional operator - misc additions   ifeqeqx 23954
19.3.3.6  Indexed union - misc additions   iuneq12daf 23960
19.3.3.7  Disjointness - misc additions   cbvdisjf 23968
19.3.4  Relations and Functions   dfrel4 23987
19.3.4.1  Relations - misc additions   dfrel4 23987
19.3.4.2  Functions - misc additions   fdmrn 23992
19.3.4.3  Isomorphisms - misc. add.   gtiso 24041
19.3.4.4  Disjointness (additional proof requiring functions)   disjdsct 24043
19.3.4.5  First and second members of an ordered pair - misc additions   df1stres 24044
19.3.4.6  Supremum - misc additions   supssd 24051
19.3.4.7  Countable Sets   nnct 24052
19.3.5  Real and Complex Numbers   addeq0 24067
19.3.5.2  Ordering on reals - misc additions   lt2addrd 24068
19.3.5.3  Extended reals - misc additions   xgepnf 24069
19.3.5.4  Real number intervals - misc additions   icossicc 24082
19.3.5.5  Finite intervals of integers - misc additions   fzssnn 24100
19.3.5.6  Half-open integer ranges - misc additions   iundisjfi 24105
19.3.5.7  The ` # ` (finite set size) function - misc additions   hashresfn 24109
19.3.5.8  The greatest common divisor operator - misc. add   numdenneg 24113
19.3.5.9  Integers   ltesubnnd 24115
19.3.5.10  Division in the extended real number system   cxdiv 24116
19.3.6  Structure builders   ress0g 24135
19.3.6.1  Structure builder restriction operator   ress0g 24135
19.3.6.2  Posets   tospos 24139
19.3.6.3  Complete lattices   clatp0ex 24146
19.3.6.4  Extended reals Structure - misc additions   ax-xrssca 24148
19.3.6.5  The extended non-negative real numbers monoid   xrge0base 24160
19.3.7  Algebra   sumpr 24171
19.3.7.1  Finitely supported group sums - misc additions   sumpr 24171
19.3.7.2  Rings - misc additions   dvrdir 24179
19.3.7.3  Ordered groups   cogrp 24184
19.3.7.4  Ordered fields   cofld 24186
19.3.7.5  The Archimedean property for generic algebraic structures   cinftm 24199
19.3.7.6  Ring homomorphisms - misc additions   rhmdvdsr 24209
19.3.7.7  The ring of integers   zzsbase 24216
19.3.7.8  The ordered field of reals   rebase 24222
19.3.8  Topology   cmetid 24234
19.3.8.1  Pseudometrics   cmetid 24234
19.3.8.2  Continuity - misc additions   hauseqcn 24246
19.3.8.3  Topology of the closed unit   unitsscn 24247
19.3.8.4  Topology of ` ( RR X. RR ) `   unicls 24254
19.3.8.5  Order topology - misc. additions   cnvordtrestixx 24264
19.3.8.6  Continuity in topological spaces - misc. additions   mndpluscn 24265
19.3.8.7  Topology of the extended non-negative real numbers monoid   xrge0hmph 24271
19.3.8.8  Limits - misc additions   lmlim 24286
19.3.9  Uniform Stuctures and Spaces   chcmp 24293
19.3.9.1  Hausdorff Completion   chcmp 24293
19.3.10  Topology and algebraic structures   zzsnm 24295
19.3.10.1  The norm on the ring of the integer numbers   zzsnm 24295
19.3.10.2  The complete ordered field of the real numbers   recms 24296
19.3.10.3  Topological ` ZZ ` -modules   zlm0 24299
19.3.10.4  The canonical embedding of the rational numbers into a division ring   cqqh 24309
19.3.10.5  The canonical embedding of ` RR ` into a complete ordered field   crrh 24330
19.3.10.6  Embedding into ` RR* `   cxrh 24335
19.3.10.7  Canonical embeddings into ` RR `   zrhre 24338
19.3.11  Real and complex functions   clogb 24341
19.3.11.1  Logarithm laws generalized to an arbitrary base - logb   clogb 24341
19.3.11.2  Indicator Functions   cind 24361
19.3.11.3  Extended sum   cesum 24377
19.3.12  Mixed Function/Constant operation   cofc 24431
19.3.13  Abstract measure   csiga 24443
19.3.13.1  Sigma-Algebra   csiga 24443
19.3.13.2  Generated Sigma-Algebra   csigagen 24474
19.3.13.3  The Borel algebra on the real numbers   cbrsiga 24488
19.3.13.4  Product Sigma-Algebra   csx 24495
19.3.13.5  Measures   cmeas 24502
19.3.13.6  The counting measure   cntmeas 24533
19.3.13.7  The Lebesgue measure - misc additions   volss 24536
19.3.13.8  The 'almost everywhere' relation   cae 24541
19.3.13.9  Measurable functions   cmbfm 24553
19.3.13.10  Borel Algebra on ` ( RR X. RR ) `   br2base 24572
19.3.14  Integration   itgeq12dv 24594
19.3.14.1  Lebesgue integral - misc additions   itgeq12dv 24594
19.3.14.2  Bochner integral   citgm 24595
19.3.15  Probability   cprb 24618
19.3.15.1  Probability Theory   cprb 24618
19.3.15.2  Conditional Probabilities   ccprob 24642
19.3.15.3  Real Valued Random Variables   crrv 24651
19.3.15.4  Preimage set mapping operator   corvc 24666
19.3.15.5  Distribution Functions   orvcelval 24679
19.3.15.6  Cumulative Distribution Functions   orvclteel 24683
19.3.15.7  Probabilities - example   coinfliplem 24689
19.3.15.8  Bertrand's Ballot Problem   ballotlemoex 24696
19.4  Mathbox for Mario Carneiro
19.4.1  Miscellaneous stuff   quartfull 24749
19.4.2  Zeta function   czeta 24750
19.4.3  Gamma function   clgam 24753
19.4.4  Derangements and the Subfactorial   deranglem 24805
19.4.5  The Erdős-Szekeres theorem   erdszelem1 24830
19.4.6  The Kuratowski closure-complement theorem   kur14lem1 24845
19.4.7  Retracts and sections   cretr 24856
19.4.8  Path-connected and simply connected spaces   cpcon 24859
19.4.9  Covering maps   ccvm 24895
19.4.10  Normal numbers   snmlff 24969
19.4.11  Godel-sets of formulas   cgoe 24973
19.4.12  Models of ZF   cgze 25001
19.4.13  Splitting fields   citr 25015
19.4.14  p-adic number fields   czr 25031
19.5  Mathbox for Paul Chapman
19.5.1  Group homomorphism and isomorphism   ghomgrpilem1 25049
19.5.2  Real and complex numbers (cont.)   climuzcnv 25061
19.5.3  Miscellaneous theorems   elfzm12 25065
19.6  Mathbox for Drahflow
19.7  Mathbox for Scott Fenton
19.7.1  ZFC Axioms in primitive form   axextprim 25103
19.7.2  Untangled classes   untelirr 25110
19.7.3  Extra propositional calculus theorems   3orel1 25117
19.7.4  Misc. Useful Theorems   nepss 25128
19.7.5  Properties of reals and complexes   sqdivzi 25137
19.7.6  Product sequences   prodf 25168
19.7.7  Non-trivial convergence   ntrivcvg 25178
19.7.8  Complex products   cprod 25184
19.7.9  Finite products   fprod 25220
19.7.10  Infinite products   iprodclim 25264
19.7.11  Falling and Rising Factorial   cfallfac 25273
19.7.12  Factorial limits   faclimlem1 25310
19.7.13  Greatest common divisor and divisibility   pdivsq 25316
19.7.14  Properties of relationships   brtp 25320
19.7.15  Properties of functions and mappings   funpsstri 25335
19.7.16  Epsilon induction   setinds 25348
19.7.17  Ordinal numbers   elpotr 25351
19.7.18  Defined equality axioms   axextdfeq 25368
19.7.19  Hypothesis builders   hbntg 25376
19.7.20  The Predecessor Class   cpred 25381
19.7.21  (Trans)finite Recursion Theorems   tfisg 25418
19.7.22  Well-founded induction   tz6.26 25419
19.7.23  Transitive closure under a relationship   ctrpred 25434
19.7.24  Founded Induction   frmin 25456
19.7.25  Ordering Ordinal Sequences   orderseqlem 25466
19.7.26  Well-founded recursion   wfr3g 25469
19.7.27  Transfinite recursion via Well-founded recursion   tfrALTlem 25490
19.7.28  Founded Recursion   frr3g 25494
19.7.29  Surreal Numbers   csur 25508
19.7.30  Surreal Numbers: Ordering   sltsolem1 25536
19.7.31  Surreal Numbers: Birthday Function   bdayfo 25543
19.7.32  Surreal Numbers: Density   fvnobday 25550
19.7.33  Surreal Numbers: Density   nodenselem3 25551
19.7.34  Surreal Numbers: Upper and Lower Bounds   nobndlem1 25560
19.7.35  Surreal Numbers: Full-Eta Property   nofulllem1 25570
19.7.36  Symmetric difference   csymdif 25575
19.7.37  Quantifier-free definitions   ctxp 25587
19.7.38  Alternate ordered pairs   caltop 25705
19.7.39  Tarskian geometry   cee 25731
19.7.40  Tarski's axioms for geometry   axdimuniq 25756
19.7.41  Congruence properties   cofs 25820
19.7.42  Betweenness properties   btwntriv2 25850
19.7.43  Segment Transportation   ctransport 25867
19.7.44  Properties relating betweenness and congruence   cifs 25873
19.7.45  Connectivity of betweenness   btwnconn1lem1 25925
19.7.46  Segment less than or equal to   csegle 25944
19.7.47  Outside of relationship   coutsideof 25957
19.7.48  Lines and Rays   cline2 25972
19.7.49  Bernoulli polynomials and sums of k-th powers   cbp 25996
19.7.50  Rank theorems   rankung 26011
19.7.51  Hereditarily Finite Sets   chf 26017
19.8  Mathbox for Anthony Hart
19.8.1  Propositional Calculus   tb-ax1 26032
19.8.2  Predicate Calculus   quantriv 26054
19.8.3  Misc. Single Axiom Systems   meran1 26065
19.8.4  Connective Symmetry   negsym1 26071
19.9  Mathbox for Chen-Pang He
19.9.1  Ordinal topology   ontopbas 26082
19.10  Mathbox for Jeff Hoffman
19.10.1  Inferences for finite induction on generic function values   fveleq 26105
19.10.2  gdc.mm   nnssi2 26109
19.11  Mathbox for Wolf Lammen
19.12  Mathbox for Brendan Leahy
19.13  Mathbox for Jeff Hankins
19.13.1  Miscellany   a1i13 26188
19.13.2  Basic topological facts   topbnd 26217
19.13.3  Topology of the real numbers   ivthALT 26228
19.13.4  Refinements   cfne 26229
19.13.5  Neighborhood bases determine topologies   neibastop1 26278
19.13.6  Lattice structure of topologies   topmtcl 26282
19.13.7  Filter bases   fgmin 26289
19.13.8  Directed sets, nets   tailfval 26291
19.14  Mathbox for Jeff Madsen
19.14.1  Logic and set theory   anim12da 26302
19.14.2  Real and complex numbers; integers   filbcmb 26332
19.14.3  Sequences and sums   sdclem2 26336
19.14.4  Topology   subspopn 26348
19.14.5  Metric spaces   metf1o 26351
19.14.6  Continuous maps and homeomorphisms   constcncf 26358
19.14.7  Boundedness   ctotbnd 26365
19.14.8  Isometries   cismty 26397
19.14.9  Heine-Borel Theorem   heibor1lem 26408
19.14.10  Banach Fixed Point Theorem   bfplem1 26421
19.14.11  Euclidean space   crrn 26424
19.14.12  Intervals (continued)   ismrer1 26437
19.14.13  Groups and related structures   exidcl 26441
19.14.14  Rings   rngonegcl 26451
19.14.15  Ring homomorphisms   crnghom 26466
19.14.16  Commutative rings   ccring 26495
19.14.17  Ideals   cidl 26507
19.14.18  Prime rings and integral domains   cprrng 26546
19.14.19  Ideal generators   cigen 26559
19.15  Mathbox for Rodolfo Medina
19.15.1  Partitions   prtlem60 26578
19.16  Mathbox for Stefan O'Rear
19.16.1  Additional elementary logic and set theory   nelss 26622
19.16.2  Additional theory of functions   fninfp 26625
19.16.3  Extensions beyond function theory   gsumvsmul 26635
19.16.4  Additional topology   elrfi 26638
19.16.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26642
19.16.6  Algebraic closure systems   cnacs 26646
19.16.7  Miscellanea 1. Map utilities   constmap 26657
19.16.8  Miscellanea for polynomials   ofmpteq 26666
19.16.9  Multivariate polynomials over the integers   cmzpcl 26668
19.16.10  Miscellanea for Diophantine sets 1   coeq0 26700
19.16.11  Diophantine sets 1: definitions   cdioph 26703
19.16.12  Diophantine sets 2 miscellanea   ellz1 26715
19.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26721
19.16.14  Diophantine sets 3: construction   diophrex 26724
19.16.15  Diophantine sets 4 miscellanea   2sbcrex 26733
19.16.16  Diophantine sets 4: Quantification   rexrabdioph 26744
19.16.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26751
19.16.18  Diophantine sets 6 miscellanea   fz1ssnn 26761
19.16.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26762
19.16.20  Pigeonhole Principle and cardinality helpers   fphpd 26767
19.16.21  A non-closed set of reals is infinite   rencldnfilem 26771
19.16.22  Miscellanea for Lagrange's theorem   icodiamlt 26773
19.16.23  Lagrange's rational approximation theorem   irrapxlem1 26775
19.16.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26782
19.16.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26789
19.16.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26831
19.16.27  Logarithm laws generalized to an arbitrary base   reglogcl 26843
19.16.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26851
19.16.29  X and Y sequences 1: Definition and recurrence laws   crmx 26853
19.16.30  Ordering and induction lemmas for the integers   monotuz 26894
19.16.31  X and Y sequences 2: Order properties   rmxypos 26902
19.16.32  Congruential equations   congtr 26920
19.16.33  Alternating congruential equations   acongid 26930
19.16.34  Additional theorems on integer divisibility   bezoutr 26940
19.16.35  X and Y sequences 3: Divisibility properties   jm2.18 26949
19.16.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26966
19.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26976
19.16.38  Uncategorized stuff not associated with a major project   setindtr 26985
19.16.39  More equivalents of the Axiom of Choice   axac10 26994
19.16.40  Finitely generated left modules   clfig 27033
19.16.41  Noetherian left modules I   clnm 27041
19.16.42  Addenda for structure powers   pwssplit0 27055
19.16.43  Direct sum of left modules   cdsmm 27065
19.16.44  Free modules   cfrlm 27080
19.16.45  Every set admits a group structure iff choice   unxpwdom3 27124
19.16.46  Independent sets and families   clindf 27142
19.16.47  Characterization of free modules   lmimlbs 27174
19.16.48  Noetherian rings and left modules II   clnr 27181
19.16.49  Hilbert's Basis Theorem   cldgis 27193
19.16.50  Additional material on polynomials [DEPRECATED]   cmnc 27203
19.16.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27213
19.16.52  Algebraic integers I   citgo 27230
19.16.53  Finite cardinality [SO]   en1uniel 27248
19.16.54  Words in monoids and ordered group sum   issubmd 27251
19.16.55  Transpositions in the symmetric group   cpmtr 27252
19.16.56  The sign of a permutation   cpsgn 27282
19.16.57  The matrix algebra   cmmul 27307
19.16.58  The determinant   cmdat 27351
19.16.59  Endomorphism algebra   cmend 27357
19.16.60  Subfields   csdrg 27371
19.16.61  Cyclic groups and order   idomrootle 27379
19.16.62  Cyclotomic polynomials   ccytp 27389
19.16.63  Miscellaneous topology   fgraphopab 27397
19.17  Mathbox for Steve Rodriguez
19.17.1  Miscellanea   iso0 27404
19.17.2  Function operations   caofcan 27408
19.17.3  Calculus   lhe4.4ex1a 27414
19.18  Mathbox for Andrew Salmon
19.18.1  Principia Mathematica * 10   pm10.12 27421
19.18.2  Principia Mathematica * 11   2alanimi 27435
19.18.3  Predicate Calculus   sbeqal1 27465
19.18.4  Principia Mathematica * 13 and * 14   pm13.13a 27475
19.18.5  Set Theory   elnev 27506
19.18.6  Arithmetic   addcomgi 27528
19.18.7  Geometry   cplusr 27529
19.19  Mathbox for Glauco Siliprandi
19.19.1  Miscellanea   ssrexf 27551
19.19.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27577
19.19.3  Limits   clim1fr1 27594
19.19.4  Derivatives   dvsinexp 27607
19.19.5  Integrals   ioovolcl 27609
19.19.6  Stone Weierstrass theorem - real version   stoweidlem1 27617
19.19.7  Wallis' product for π   wallispilem1 27681
19.19.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27690
19.20  Mathbox for Saveliy Skresanov
19.20.1  Ceva's theorem   sigarval 27707
19.21  Mathbox for Jarvin Udandy
19.22  Mathbox for Alexander van der Vekens
19.22.1  Double restricted existential uniqueness   r19.32 27812
19.22.1.1  Restricted quantification (extension)   r19.32 27812
19.22.1.2  The empty set (extension)   raaan2 27820
19.22.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 27821
19.22.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 27826
19.22.2  Alternative definitions of function's and operation's values   wdfat 27838
19.22.2.1  Restricted quantification (extension)   ralbinrald 27844
19.22.2.2  The universal class (extension)   nvelim 27845
19.22.2.3  Introduce the Axiom of Power Sets (extension)   alneu 27846
19.22.2.4  Relations (extension)   sbcrel 27848
19.22.2.5  Functions (extension)   sbcfun 27854
19.22.2.6  Predicate "defined at"   dfateq12d 27860
19.22.2.7  Alternative definition of the value of a function   dfafv2 27863
19.22.2.8  Alternative definition of the value of an operation   aoveq123d 27909
19.22.3  Auxiliary theorems for graph theory   eqneqall 27939
19.22.3.1  Negated equality and membership - extension   eqneqall 27939
19.22.3.2  "Weak deduction theorem" for set theory - extension   2if2 27941
19.22.3.3  Power classes - extension   3xpexg 27942
19.22.3.4  Unordered and ordered pairs - extension   nelprd 27943
19.22.3.5  Indexed union and intersection - extension   iunxprg 27956
19.22.3.6  Relations - extension   resisresindm 27957
19.22.3.7  Functions - extension   2f1fvneq 27958
19.22.3.8  Equinumerosity - extension   resfnfinfin 27966
19.22.3.9  Subtraction - extension   cnm1cn 27968
19.22.3.10  Multiplication - extension   kcnktkm1cn 27969
19.22.3.11  Ordering on reals (cont.) - extension   leaddsuble 27970
19.22.3.12  Nonnegative integers (as a subset of complex numbers) - extension   0mnnnnn0 27971
19.22.3.13  Finite intervals of integers - extension   ssfz12 27976
19.22.3.14  Half-open integer ranges (extension)   fzo0ss1 27985
19.22.3.15  The ` # ` (finite set size) function - extension   hashimarn 27994
19.22.3.16  Words over a set - extension   iswrd0i 27999
19.22.3.17  Words over a set - extension (subwords of subwords)   swrd0swrd 28009
19.22.3.18  Words over a set - extension (subwords of concatenations)   swrdccat3a0 28015
19.22.4  Graph theory   uhgraedgrnv 28032
19.22.4.1  Undirected hypergraphs   uhgraedgrnv 28032
19.22.4.2  Undirected simple graphs   usisuhgra 28033
19.22.4.3  Neighbors, complete graphs and universal vertices   nbfiusgrafi 28034
19.22.4.4  Walks, Paths and Cycles   usgra2pthspth 28035
19.22.4.5  Walks/paths of length 2 as ordered triples   c2wlkot 28051
19.22.4.6  Vertex Degree   usgfidegfi 28090
19.22.4.7  Friendship graphs   cfrgra 28092
19.23  Mathbox for David A. Wheeler
19.23.1  Natural deduction   19.8ad 28174
19.23.2  Greater than, greater than or equal to.   cge-real 28177
19.23.3  Hyperbolic trig functions   csinh 28187
19.23.4  Reciprocal trig functions (sec, csc, cot)   csec 28198
19.23.5  Identities for "if"   ifnmfalse 28220
19.23.6  Not-member-of   AnelBC 28221
19.23.7  Decimal point   cdp2 28222
19.23.8  Signum (sgn or sign) function   csgn 28230
19.23.9  Ceiling function   ccei 28240
19.23.10  Logarithms generalized to arbitrary base using ` logb `   ene0 28244
19.23.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 28247
19.23.12  Miscellaneous   5m4e1 28249
19.24  Mathbox for Alan Sare
19.24.2  Supplementary unification deductions   biimp 28278
19.24.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 28294
19.24.4  What is Virtual Deduction?   wvd1 28369
19.24.5  Virtual Deduction Theorems   df-vd1 28370
19.24.6  Theorems proved using virtual deduction   trsspwALT 28640
19.24.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 28667
19.24.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 28734
19.24.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 28738
19.24.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 28745
19.24.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 28748
19.25  Mathbox for Jonathan Ben-Naim
19.25.1  First order logic and set theory   bnj170 28768
19.25.2  Well founded induction and recursion   bnj110 28935
19.25.3  The existence of a minimal element in certain classes   bnj69 29085
19.25.4  Well-founded induction   bnj1204 29087
19.25.5  Well-founded recursion, part 1 of 3   bnj60 29137
19.25.6  Well-founded recursion, part 2 of 3   bnj1500 29143
19.25.7  Well-founded recursion, part 3 of 3   bnj1522 29147
19.26  Mathbox for Norm Megill
19.26.1  Experiments to study ax-7 unbundling   ax-7v 29148
19.26.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)   ax-7v 29148
19.26.1.2  Theorems derived from ax-7 (suffix OLD7)   ax-7OLD7 29362
19.26.2  Miscellanea   cnaddcom 29454
19.26.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 29457
19.26.4  Functionals and kernels of a left vector space (or module)   clfn 29540
19.26.5  Opposite rings and dual vector spaces   cld 29606
19.26.6  Ortholattices and orthomodular lattices   cops 29655
19.26.7  Atomic lattices with covering property   ccvr 29745
19.26.8  Hilbert lattices   chlt 29833
19.26.9  Projective geometries based on Hilbert lattices   clln 29973
19.26.10  Construction of a vector space from a Hilbert lattice   cdlema1N 30273
19.26.11  Construction of involution and inner product from a Hilbert lattice   clpoN 31963

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32447
 Copyright terms: Public domain < Wrap  Next >