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

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page:  Detailed Table of Contents  Page List

Table of Contents Summary
PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Conventions
      1.3  Propositional calculus
      1.4  Other axiomatizations of classical propositional calculus
      1.5  Predicate calculus mostly without distinct variables
      1.6  Predicate calculus with distinct variables
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarksi-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Elementary real and complex functions
      5.8  Elementary limits and convergence
      5.9  Elementary trigonometry
      5.10  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  EXTENSIBLE STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Presets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Abelian groups
      10.4  Rings
      10.5  Division rings and Fields
      10.6  Left Modules
      10.7  Vector Spaces
      10.8  Ideals
      10.9  Associative algebras
      10.10  Abstract Multivariate Polynomials
      10.11  The complex numbers as an extensible structure
      10.12  Hilbert spaces
PART 11  BASIC TOPOLOGY
      11.1  Topology
      11.2  Filters and filter bases
      11.3  Metric spaces
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
      12.2  Integrals
      12.3  Derivatives
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
      13.2  Sequences and series
      13.3  Basic trigonometry
      13.4  Basic number theory
PART 14  MISCELLANEA
      14.1  Definitional Examples
      14.2  Natural deduction examples
      14.3  Humor
      14.4  (Future - to be reviewed and classified)
PART 15  DEPRECATED SECTIONS
      15.1  Additional material on Group theory
      15.2  Additional material on Rings and Fields
      15.3  Complex vector spaces
      15.4  Normed complex vector spaces
      15.5  Operators on complex vector spaces
      15.6  Inner product (pre-Hilbert) spaces
      15.7  Complex Banach spaces
      15.8  Complex Hilbert spaces
      15.9  Hilbert Space Explorer
PART 16  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      16.1  Mathboxes for user contributions
      16.2  Mathbox for Stefan Allan
      16.3  Mathbox for Thierry Arnoux
      16.4  Mathbox for Mario Carneiro
      16.5  Mathbox for Paul Chapman
      16.6  Mathbox for Drahflow
      16.7  Mathbox for Scott Fenton
      16.8  Mathbox for Anthony Hart
      16.9  Mathbox for Chen-Pang He
      16.10  Mathbox for Jeff Hoffman
      16.11  Mathbox for Wolf Lammen
      16.12  Mathbox for Frédéric Liné
      16.13  Mathbox for Jeff Hankins
      16.14  Mathbox for Jeff Madsen
      16.15  Mathbox for Rodolfo Medina
      16.16  Mathbox for Stefan O'Rear
      16.17  Mathbox for Steve Rodriguez
      16.18  Mathbox for Andrew Salmon
      16.19  Mathbox for Glauco Siliprandi
      16.20  Mathbox for Jarvin Udandy
      16.21  Mathbox for David A. Wheeler
      16.22  Mathbox for Alan Sare
      16.23  Mathbox for Jonathan Ben-Naim
      16.24  Mathbox for Norm Megill

Detailed Table of Contents
PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
            1.1.1  Inferences for assisting proof development   dummylink 1
      1.2  Conventions
      1.3  Propositional calculus
            1.3.1  Recursively define primitive wffs for propositional calculus   wn 5
            1.3.2  The axioms of propositional calculus   ax-1 7
            1.3.3  Logical implication   mp2b 11
            1.3.4  Logical negation   con4d 99
            1.3.5  Logical equivalence   wb 178
            1.3.6  Logical disjunction and conjunction   wo 359
            1.3.7  Miscellaneous theorems of propositional calculus   pm5.21nd 873
            1.3.8  Abbreviated conjunction and disjunction of three wff's   w3o 938
            1.3.9  Logical 'nand' (Sheffer stroke)   wnan 1292
            1.3.10  Logical 'xor'   wxo 1300
            1.3.11  True and false constants   wtru 1312
            1.3.12  Truth tables   truantru 1332
            1.3.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1358
            1.3.14  Half-adders and full adders in propositional calculus   whad 1374
      1.4  Other axiomatizations of classical propositional calculus
            1.4.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1400
            1.4.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1418
            1.4.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1429
            1.4.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1435
            1.4.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1454
            1.4.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1458
            1.4.7  Deriving the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1473
            1.4.8  Deriving the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1496
            1.4.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1509
            1.4.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1528
      1.5  Predicate calculus mostly without distinct variables
            1.5.1  "Pure" (equality-free) predicate calculus axioms ax-5, ax-6, ax-7, ax-gen   wal 1532
            1.5.2  Introduce equality axioms ax-8, ax-11, ax-13, and ax-14   cv 1618
            1.5.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1628
            1.5.4  Introduce equality axioms ax-9v and ax-12   ax-9v 1632
            1.5.5  Derive ax-12o from ax-12   ax12o10lem1 1635
            1.5.6  Derive ax-10   ax10lem16 1665
            1.5.7  Derive ax-9 from the weaker version ax-9v   ax9 1683
            1.5.8  Introduce Axiom of Existence ax-9   ax-9 1684
            1.5.9  Derive ax-4, ax-5o, and ax-6o   ax4 1691
            1.5.10  "Pure" predicate calculus including ax-4, without distinct variables   a4i 1699
            1.5.11  Equality theorems without distinct variables   ax9o 1814
            1.5.12  Axioms ax-10 and ax-11   ax10o 1835
            1.5.13  Substitution (without distinct variables)   wsb 1883
            1.5.14  Theorems using axiom ax-11   equs5a 1912
      1.6  Predicate calculus with distinct variables
            1.6.1  Derive the axiom of distinct variables ax-16   a4imv 1923
            1.6.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1940
            1.6.3  Theorems without distinct variables that use axiom ax-11o   ax11b 1943
            1.6.4  Predicate calculus with distinct variables (cont.)   ax11v 1991
            1.6.5  More substitution theorems   equsb3lem 2064
            1.6.6  Existential uniqueness   weu 2117
      1.7  Other axiomatizations related to classical predicate calculus
            1.7.1  Predicate calculus with all distinct variables   ax-7d 2207
            1.7.2  Aristotelian logic: Assertic syllogisms   barbara 2213
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 2237
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2242
            2.1.3  Class form not-free predicate   wnfc 2379
            2.1.4  Negated equality and membership   wne 2419
            2.1.5  Restricted quantification   wral 2516
            2.1.6  The universal class   cvv 2740
            2.1.7  Conditional equality (experimental)   wcdeq 2918
            2.1.8  Russell's Paradox   ru 2934
            2.1.9  Proper substitution of classes for sets   wsbc 2935
            2.1.10  Proper substitution of classes for sets into classes   csb 3023
            2.1.11  Define basic set operations and relations   cdif 3091
            2.1.12  Subclasses and subsets   df-ss 3108
            2.1.13  The difference, union, and intersection of two classes   difeq1 3229
            2.1.14  The empty set   c0 3397
            2.1.15  "Weak deduction theorem" for set theory   cif 3506
            2.1.16  Power classes   cpw 3566
            2.1.17  Unordered and ordered pairs   csn 3581
            2.1.18  The union of a class   cuni 3768
            2.1.19  The intersection of a class   cint 3803
            2.1.20  Indexed union and intersection   ciun 3846
            2.1.21  Disjointness   wdisj 3934
            2.1.22  Binary relations   wbr 3963
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4016
            2.1.24  Transitive classes   wtr 4053
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4071
            2.2.2  Derive the Axiom of Separation   axsep 4080
            2.2.3  Derive the Null Set Axiom   zfnuleu 4086
            2.2.4  Theorems requiring subset and intersection existence   nalset 4091
            2.2.5  Theorems requiring empty set existence   class2set 4116
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4126
            2.3.2  Derive the Axiom of Pairing   zfpair 4150
            2.3.3  Ordered pair theorem   opnz 4179
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4208
            2.3.5  Power class of union and intersection   pwin 4234
            2.3.6  Epsilon and identity relations   cep 4240
            2.3.7  Partial and complete ordering   wpo 4249
            2.3.8  Founded and well-ordering relations   wfr 4286
            2.3.9  Ordinals   word 4328
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4449
            2.4.2  Ordinals (continued)   ordon 4511
            2.4.3  Transfinite induction   tfi 4581
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4593
            2.4.5  Peano's postulates   peano1 4612
            2.4.6  Finite induction (for finite ordinals)   find 4618
            2.4.7  Functions and relations   cxp 4624
            2.4.8  Operations   co 5757
            2.4.9  "Maps to" notation   elmpt2cl 5960
            2.4.10  Function operation   cof 5975
            2.4.11  First and second members of an ordered pair   c1st 6019
            2.4.12  Function transposition   ctpos 6132
            2.4.13  Curry and uncurry   ccur 6171
            2.4.14  Proper subset relation   crpss 6175
            2.4.15  Definite description binder (inverted iota)   cio 6188
            2.4.16  Cantor's Theorem   canth 6225
            2.4.17  Undefined values and restricted iota (description binder)   cund 6227
            2.4.18  Functions on ordinals; strictly monotone ordinal functions   iunon 6288
            2.4.19  "Strong" transfinite recursion   crecs 6320
            2.4.20  Recursive definition generator   crdg 6355
            2.4.21  Finite recursion   frfnom 6380
            2.4.22  Abian's "most fundamental" fixed point theorem   abianfplem 6403
            2.4.23  Ordinal arithmetic   c1o 6405
            2.4.24  Natural number arithmetic   nna0 6535
            2.4.25  Equivalence relations and classes   wer 6590
            2.4.26  The mapping operation   cmap 6705
            2.4.27  Infinite Cartesian products   cixp 6750
            2.4.28  Equinumerosity   cen 6793
            2.4.29  Schroeder-Bernstein Theorem   sbthlem1 6904
            2.4.30  Equinumerosity (cont.)   xpf1o 6956
            2.4.31  Pigeonhole Principle   phplem1 6973
            2.4.32  Finite sets   onomeneq 6983
            2.4.33  Finite intersections   cfi 7097
            2.4.34  Hall's marriage theorem   marypha1lem 7119
            2.4.35  Supremum   csup 7126
            2.4.36  Ordinal isomorphism, Hartog's theorem   coi 7157
            2.4.37  Hartogs function, order types, weak dominance   char 7203
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7239
            2.5.2  Axiom of Infinity equivalents   inf0 7255
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7272
            2.6.2  Existence of omega (the set of natural numbers)   omex 7277
            2.6.3  Cantor normal form   ccnf 7295
            2.6.4  Transitive closure   trcl 7343
            2.6.5  Rank   cr1 7367
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7488
            2.6.7  Cardinal numbers   ccrd 7501
            2.6.8  Axiom of Choice equivalents   wac 7675
            2.6.9  Cardinal number arithmetic   ccda 7726
            2.6.10  The Ackermann bijection   ackbij2lem1 7778
            2.6.11  Cofinality (without Axiom of Choice)   cflem 7805
            2.6.12  Eight inequivalent definitions of finite set   sornom 7836
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 7975
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 8018
            3.2.2  AC equivalents: well ordering, Zorn's lemma   numthcor 8054
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8101
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8129
            3.2.5  Cofinality using Axiom of Choice   alephreg 8137
      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 8237
            4.1.2  Weak universes   cwun 8255
            4.1.3  Tarski's classes   ctsk 8303
            4.1.4  Grothendieck's universes   cgru 8345
      4.2  ZFC Set Theory plus the Tarksi-Grothendieck Axiom
            4.2.1  Introduce the Tarksi-Grothendieck Axiom   ax-groth 8378
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8381
            4.2.3  Tarski map function   ctskm 8392
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 8399
            5.1.2  Final derivation of real and complex number postulates   axaddf 8700
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8726
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 8751
            5.2.2  Infinity and the extended real number system   cpnf 8797
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 8827
            5.2.4  Ordering on reals   lttr 8832
            5.2.5  Initial properties of the complex numbers   mul12 8911
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 8958
            5.3.2  Subtraction   cmin 8970
            5.3.3  Multiplication   muladd 9145
            5.3.4  Ordering on reals (cont.)   gt0ne0 9172
            5.3.5  Reciprocals   ixi 9330
            5.3.6  Division   cdiv 9356
            5.3.7  Ordering on reals (cont.)   elimgt0 9525
            5.3.8  Completeness Axiom and Suprema   fimaxre 9634
            5.3.9  Imaginary and complex number properties   inelr 9669
            5.3.10  Function operation analogue theorems   ofsubeq0 9676
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9679
            5.4.2  Principle of mathematical induction   nnind 9697
            5.4.3  Decimal representation of numbers   c2 9728
            5.4.4  Some properties of specific numbers   0p1e1 9772
            5.4.5  The Archimedean property   nnunb 9893
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 9897
            5.4.7  Integers (as a subset of complex numbers)   cz 9956
            5.4.8  Decimal arithmetic   cdc 10056
            5.4.9  Upper partititions of integers   cuz 10162
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10243
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10248
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10274
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10286
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10381
            5.5.3  Supremum on the extended reals   xrsupexmnf 10554
            5.5.4  Real number intervals   cioo 10587
            5.5.5  Finite intervals of integers   cfz 10713
            5.5.6  Half-open integer ranges   cfzo 10801
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 10855
            5.6.2  The modulo (remainder) operation   cmo 10904
            5.6.3  The infinite sequence builder "seq"   om2uz0i 10941
            5.6.4  Integer powers   cexp 11035
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11213
            5.6.6  Factorial function   cfa 11219
            5.6.7  The binomial coefficient operation   cbc 11246
            5.6.8  The ` # ` (finite set size) function   chash 11268
            5.6.9  Words over a set   cword 11333
            5.6.10  Longer string literals   cs2 11421
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11491
            5.7.2  Real and imaginary parts; conjugate   ccj 11511
            5.7.3  Square root; absolute value   csqr 11648
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 11874
            5.8.2  Limits   cli 11888
            5.8.3  Finite and infinite sums   csu 12088
            5.8.4  The binomial theorem   binomlem 12217
            5.8.5  Infinite sums (cont.)   isumshft 12225
            5.8.6  Miscellaneous converging and diverging sequences   divrcnv 12238
            5.8.7  Arithmetic series   arisum 12245
            5.8.8  Geometric series   expcnv 12249
            5.8.9  Ratio test for infinite series convergence   cvgrat 12266
            5.8.10  Mertens' theorem   mertenslem1 12267
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12270
            5.9.2  _e is irrational   eirrlem 12409
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12414
            5.10.2  The reals are uncountable   rpnnen2lem1 12420
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12453
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12456
            6.1.3  The divides relation   cdivides 12458
            6.1.4  The division algorithm   divalglem0 12519
            6.1.5  Bit sequences   cbits 12537
            6.1.6  The greatest common divisor operator   cgcd 12612
            6.1.7  Bézout's identity   bezoutlem1 12644
            6.1.8  Algorithms   nn0seqcvgd 12667
            6.1.9  Euclid's Algorithm   eucalgval2 12678
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 12685
            6.2.2  Properties of the canonical representation of a rational   cnumer 12731
            6.2.3  Euler's theorem   codz 12758
            6.2.4  Pythagorean Triples   coprimeprodsq 12789
            6.2.5  The prime count function   cpc 12816
            6.2.6  Pocklington's theorem   prmpwdvds 12878
            6.2.7  Infinite primes theorem   unbenlem 12882
            6.2.8  Sum of prime reciprocals   prmreclem1 12890
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 12897
            6.2.10  Lagrange's four-square theorem   cgz 12903
            6.2.11  Van der Waerden's theorem   cvdwa 12939
            6.2.12  Ramsey's theorem   cram 12973
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13005
            6.2.14  Specific prime numbers   4nprm 13033
            6.2.15  Very large primes   1259lem1 13056
PART 7  EXTENSIBLE STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13071
            7.1.2  Slot definitions   cplusg 13135
            7.1.3  Definition of the structure product   crest 13252
            7.1.4  Definition of the structure quotient   cordt 13325
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13435
            7.2.2  Independent sets in a Moore system   mrisval 13459
            7.2.3  Algebraic closure systems   isacs 13480
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13493
            8.1.2  Opposite category   coppc 13541
            8.1.3  Monomorphisms and epimorphisms   cmon 13558
            8.1.4  Sections, inverses, isomorphisms   csect 13574
            8.1.5  Subcategories   cssc 13611
            8.1.6  Functors   cfunc 13655
            8.1.7  Full & faithful functors   cful 13703
            8.1.8  Natural transformations and the functor category   cnat 13742
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 13812
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 13834
            8.3.2  The category of categories   ccatc 13853
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 13869
            8.4.2  Functor evaluation   cevlf 13910
            8.4.3  Hom functor   chof 13949
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 14001
            9.2.2  Lattices   clat 14078
            9.2.3  The dual of an ordered set   codu 14159
            9.2.4  Subset order structures   cipo 14181
            9.2.5  Distributive lattices   latmass 14218
            9.2.6  Posets and lattices as relations   cps 14228
            9.2.7  Directed sets, nets   cdir 14277
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14288
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14340
            10.1.3  Ordered group sum operation   gsumvallem1 14375
            10.1.4  Free monoids   cfrmd 14396
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14416
            10.2.2  Subgroups and Quotient groups   csubg 14542
            10.2.3  Elementary theory of group homomorphisms   cghm 14607
            10.2.4  Isomorphisms of groups   cgim 14648
            10.2.5  Group actions   cga 14670
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 14696
            10.2.7  Centralizers and centers   ccntz 14718
            10.2.8  The opposite group   coppg 14745
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 14767
            10.2.10  Direct products   clsm 14872
            10.2.11  Free groups   cefg 14942
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15016
            10.3.2  Cyclic groups   ccyg 15091
            10.3.3  Group sum operation   gsumval3a 15116
            10.3.4  Internal direct products   cdprd 15158
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15227
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15252
            10.4.2  Definition and basic properties   crg 15264
            10.4.3  Opposite ring   coppr 15331
            10.4.4  Divisibility   cdsr 15347
            10.4.5  Ring homomorphisms   crh 15421
      10.5  Division rings and Fields
            10.5.1  Definition and basic properties   cdr 15439
            10.5.2  Subrings of a ring   csubrg 15468
            10.5.3  Absolute value (abstract algebra)   cabv 15508
            10.5.4  Star rings   cstf 15535
      10.6  Left Modules
            10.6.1  Definition and basic properties   clmod 15554
            10.6.2  Subspaces and spans in a left module   clss 15616
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 15703
            10.6.4  Subspace sum; bases for a left module   clbs 15754
      10.7  Vector Spaces
            10.7.1  Definition and basic properties   clvec 15782
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 15848
            10.8.2  Two-sided ideals and quotient rings   c2idl 15910
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 15920
            10.8.4  Nonzero rings   cnzr 15936
            10.8.5  Left regular elements. More kinds of ring   crlreg 15947
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 15977
      10.10  Abstract Multivariate Polynomials
            10.10.1  Definition and basic properties   cmps 16014
            10.10.2  Polynomial evaluation   evlslem4 16172
            10.10.3  Univariate Polynomials   cps1 16177
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16296
            10.11.2  Algebraic constructions based on the complexes   czrh 16378
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16455
            10.12.2  Orthocomplements and closed subspaces   cocv 16487
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16527
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16558
            11.1.2  TopBases for topologies   isbasisg 16612
            11.1.3  Examples of topologies   distop 16660
            11.1.4  Closure and interior   ccld 16680
            11.1.5  Neighborhoods   cnei 16761
            11.1.6  Limit points and perfect sets   clp 16793
            11.1.7  Subspace topologies   restrcl 16815
            11.1.8  Order topology   ordtbaslem 16845
            11.1.9  Limits and Continuity in topological spaces   ccn 16881
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 16961
            11.1.11  Compactness   ccmp 17040
            11.1.12  Connectedness   ccon 17064
            11.1.13  First- and second-countability   c1stc 17090
            11.1.14  Local topological properties   clly 17117
            11.1.15  Compactly generated spaces   ckgen 17155
            11.1.16  Product topologies   ctx 17182
            11.1.17  Continuous function-builders   cnmptid 17282
            11.1.18  Quotient maps and quotient topology   ckq 17311
            11.1.19  Homeomorphisms   chmeo 17371
      11.2  Filters and filter bases
            11.2.1  Filter Bases   cfbas 17445
            11.2.2  Filters   cfil 17467
            11.2.3  Ultrafilters   cufil 17521
            11.2.4  Filter limits   cfm 17555
            11.2.5  Topological groups   ctmd 17680
            11.2.6  Infinite group sum on topological groups   ctsu 17735
            11.2.7  Topological rings, fields, vector spaces   ctrg 17765
      11.3  Metric spaces
            11.3.1  Basic metric space properties   cxme 17809
            11.3.2  Metric space balls   blfval 17874
            11.3.3  Open sets of a metric space   mopnval 17911
            11.3.4  Continuity in metric spaces   metcnp3 18013
            11.3.5  Examples of metric spaces   dscmet 18022
            11.3.6  Normed algebraic structures   cnm 18026
            11.3.7  Normed space homomorphisms (bounded linear operators)   cnmo 18141
            11.3.8  Topology on the Reals   qtopbaslem 18194
            11.3.9  Topological definitions using the reals   cii 18306
            11.3.10  Path homotopy   chtpy 18392
            11.3.11  The fundamental group   cpco 18425
            11.3.12  Complex left modules   cclm 18487
            11.3.13  Complex pre-Hilbert space   ccph 18529
            11.3.14  Convergence and completeness   ccfil 18605
            11.3.15  Baire's Category Theorem   bcthlem1 18673
            11.3.16  Banach spaces and complex Hilbert spaces   ccms 18681
            11.3.17  Minimizing Vector Theorem   minveclem1 18715
            11.3.18  Projection theorem   pjthlem1 18728
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 18735
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 18749
            12.2.2  Lebesgue integration   cmbf 18896
      12.3  Derivatives
            12.3.1  Real and Complex Differentiation   climc 19139
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19324
            13.1.2  Polynomial degrees   cmdg 19366
            13.1.3  The division algorithm for univariate polynomials   cmn1 19438
            13.1.4  Elementary properties of complex polynomials   cply 19493
            13.1.5  The Division algorithm for polynomials   cquot 19597
            13.1.6  Algebraic numbers   caa 19621
            13.1.7  Liouville's approximation theorem   aalioulem1 19639
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 19659
            13.2.2  Uniform convergence   culm 19682
            13.2.3  Power series   pserval 19713
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 19746
            13.3.2  Properties of pi = 3.14159...   pilem1 19754
            13.3.3  Mapping of the exponential function   efgh 19830
            13.3.4  The natural logarithm on complex numbers   clog 19839
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20026
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20062
            13.3.7  Inverse trigonometric functions   casin 20085
            13.3.8  The Birthday Problem   log2ublem1 20169
            13.3.9  Areas in R^2   carea 20177
            13.3.10  More miscellaneous converging sequences   rlimcnp 20187
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20206
            13.3.12  Euler-Mascheroni constant   cem 20213
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20233
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20237
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20245
            13.4.4  Number-theoretical functions   ccht 20255
            13.4.5  Perfect Number Theorem   mersenne 20393
            13.4.6  Characters of Z/nZ   cdchr 20398
            13.4.7  Bertrand's postulate   bcctr 20441
            13.4.8  Legendre symbol   clgs 20460
            13.4.9  Quadratic Reciprocity   lgseisenlem1 20515
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 20529
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 20545
            13.4.12  The Prime Number Theorem   mudivsum 20606
            13.4.13  Ostrowski's theorem   abvcxp 20691
PART 14  MISCELLANEA
      14.1  Definitional Examples
      14.2  Natural deduction examples
      14.3  Humor
            14.3.1  April Fool's theorem   avril1 20761
      14.4  (Future - to be reviewed and classified)
            14.4.1  Planar incidence geometry   cplig 20767
            14.4.2  Algebra preliminaries   crpm 20772
            14.4.3  Transitive closure   ctcl 20774
PART 15  DEPRECATED SECTIONS
      15.1  Additional material on Group theory
            15.1.1  Definitions and basic properties for groups   cgr 20778
            15.1.2  Definition and basic properties of Abelian groups   cablo 20873
            15.1.3  Subgroups   csubgo 20893
            15.1.4  Operation properties   cass 20904
            15.1.5  Group-like structures   cmagm 20910
            15.1.6  Examples of Abelian groups   ablosn 20939
            15.1.7  Group homomorphism and isomorphism   cghom 20949
      15.2  Additional material on Rings and Fields
            15.2.1  Definition and basic properties   crngo 20967
            15.2.2  Examples of rings   cnrngo 20995
            15.2.3  Division Rings   cdrng 20997
            15.2.4  Star Fields   csfld 21000
            15.2.5  Fields and Rings   ccm2 21002
      15.3  Complex vector spaces
            15.3.1  Definition and basic properties   cvc 21026
            15.3.2  Examples of complex vector spaces   cncvc 21064
      15.4  Normed complex vector spaces
            15.4.1  Definition and basic properties   cnv 21065
            15.4.2  Examples of normed complex vector spaces   cnnv 21170
            15.4.3  Induced metric of a normed complex vector space   imsval 21179
            15.4.4  Inner product   cdip 21198
            15.4.5  Subspaces   css 21222
      15.5  Operators on complex vector spaces
            15.5.1  Definitions and basic properties   clno 21243
      15.6  Inner product (pre-Hilbert) spaces
            15.6.1  Definition and basic properties   ccphlo 21315
            15.6.2  Examples of pre-Hilbert spaces   cncph 21322
            15.6.3  Properties of pre-Hilbert spaces   isph 21325
      15.7  Complex Banach spaces
            15.7.1  Definition and basic properties   ccbn 21366
            15.7.2  Examples of complex Banach spaces   cnbn 21373
            15.7.3  Uniform Boundedness Theorem   ubthlem1 21374
            15.7.4  Minimizing Vector Theorem   minvecolem1 21378
      15.8  Complex Hilbert spaces
            15.8.1  Definition and basic properties   chlo 21389
            15.8.2  Standard axioms for a complex Hilbert space   hlex 21402
            15.8.3  Examples of complex Hilbert spaces   cnchl 21420
            15.8.4  Subspaces   ssphl 21421
            15.8.5  Hellinger-Toeplitz Theorem   htthlem 21422
      15.9  Hilbert Space Explorer
            15.9.1  Basic Hilbert space definitions   chil 21424
            15.9.2  Preliminary ZFC lemmas   df-hnorm 21473
            15.9.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 21486
            15.9.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 21504
            15.9.5  Vector operations   hvmulex 21516
            15.9.6  Inner product postulates for a Hilbert space   ax-hfi 21583
            15.9.7  Inner product   his5 21590
            15.9.8  Norms   dfhnorm2 21626
            15.9.9  Relate Hilbert space to normed complex vector spaces   hilablo 21664
            15.9.10  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 21683
            15.9.11  Cauchy sequences and limits   hcau 21688
            15.9.12  Derivation of the completeness axiom from ZF set theory   hilmet 21698
            15.9.13  Completeness postulate for a Hilbert space   ax-hcompl 21706
            15.9.14  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 21707
            15.9.15  Subspaces   df-sh 21711
            15.9.16  Closed subspaces   df-ch 21726
            15.9.17  Orthocomplements   df-oc 21756
            15.9.18  Subspace sum, span, lattice join, lattice supremum   df-shs 21812
            15.9.19  Projection theorem   pjhthlem1 21895
            15.9.20  Projectors   df-pjh 21899
            15.9.21  Orthomodular law   omlsilem 21906
            15.9.22  Projectors (cont.)   pjhtheu2 21920
            15.9.23  Hilbert lattice operations   sh0le 21944
            15.9.24  Span (cont.) and one-dimensional subspaces   spansn0 22045
            15.9.25  Operator sum, difference, and scalar multiplication   df-hosum 22087
            15.9.26  Commutes relation for Hilbert lattice elements   df-cm 22105
            15.9.27  Foulis-Holland theorem   fh1 22140
            15.9.28  Quantum Logic Explorer axioms   qlax1i 22149
            15.9.29  Orthogonal subspaces   chscllem1 22159
            15.9.30  Orthoarguesian laws 5OA and 3OA   5oalem1 22176
            15.9.31  Projectors (cont.)   pjorthi 22191
            15.9.32  Mayet's equation E_3   mayete3i 22250
            15.9.33  Zero and identity operators   df-h0op 22253
            15.9.34  Operations on Hilbert space operators   hoaddcl 22263
            15.9.35  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 22344
            15.9.36  Linear and continuous functionals and norms   df-nmfn 22350
            15.9.37  Adjoint   df-adjh 22354
            15.9.38  Dirac bra-ket notation   df-bra 22355
            15.9.39  Positive operators   df-leop 22357
            15.9.40  Eigenvectors, eigenvalues, spectrum   df-eigvec 22358
            15.9.41  Theorems about operators and functionals   nmopval 22361
            15.9.42  Riesz lemma   riesz3i 22567
            15.9.43  Adjoints (cont.)   cnlnadjlem1 22572
            15.9.44  Quantum computation error bound theorem   unierri 22609
            15.9.45  Dirac bra-ket notation (cont.)   branmfn 22610
            15.9.46  Positive operators (cont.)   leopg 22627
            15.9.47  Projectors as operators   pjhmopi 22651
            15.9.48  States on a Hilbert lattice   df-st 22716
            15.9.49  Godowski's equation   golem1 22776
            15.9.50  Covers relation; modular pairs   df-cv 22784
            15.9.51  Atoms   df-at 22843
            15.9.52  Superposition principle   superpos 22859
            15.9.53  Atoms, exchange and covering properties, atomicity   chcv1 22860
            15.9.54  Irreducibility   chirredlem1 22895
            15.9.55  Atoms (cont.)   atcvat3i 22901
            15.9.56  Modular symmetry   mdsymlem1 22908
PART 16  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      16.1  Mathboxes for user contributions
            16.1.1  Mathbox guidelines   mathbox 22947
      16.2  Mathbox for Stefan Allan
      16.3  Mathbox for Thierry Arnoux
            16.3.1  Bertrand's Ballot Problem   ballotlemoex 22970
      16.4  Mathbox for Mario Carneiro
            16.4.1  Miscellaneous stuff   quartfull 23023
            16.4.2  Zeta function   czeta 23024
            16.4.3  Gamma function   clgam 23027
            16.4.4  Derangements and the Subfactorial   deranglem 23034
            16.4.5  The Erdős-Szekeres theorem   erdszelem1 23059
            16.4.6  The Kuratowski closure-complement theorem   kur14lem1 23074
            16.4.7  Retracts and sections   cretr 23085
            16.4.8  Path-connected and simply connected spaces   cpcon 23087
            16.4.9  Covering maps   ccvm 23123
            16.4.10  Undirected multigraphs   cumg 23197
            16.4.11  Normal numbers   snmlff 23249
            16.4.12  Godel-sets of formulas   cgoe 23253
            16.4.13  Models of ZF   cgze 23281
            16.4.14  Splitting fields   citr 23295
            16.4.15  p-adic number fields   czr 23311
      16.5  Mathbox for Paul Chapman
            16.5.1  Group homomorphism and isomorphism   ghomgrpilem1 23329
            16.5.2  Real and complex numbers (cont.)   climuzcnv 23341
            16.5.3  Miscellaneous theorems   elfzm12 23345
      16.6  Mathbox for Drahflow
      16.7  Mathbox for Scott Fenton
            16.7.1  ZFC Axioms in primitive form   axextprim 23384
            16.7.2  Untangled classes   untelirr 23391
            16.7.3  Extra propositional calculus theorems   3orel1 23398
            16.7.4  Misc. Useful Theorems   nepss 23409
            16.7.5  Properties of reals and complexes   sqdivzi 23415
            16.7.6  Greatest common divisor and divisibility   pdivsq 23438
            16.7.7  Properties of relationships   brtp 23442
            16.7.8  Properties of functions and mappings   funpsstri 23455
            16.7.9  Epsilon induction   setinds 23468
            16.7.10  Ordinal numbers   elpotr 23471
            16.7.11  Defined equality axioms   axextdfeq 23488
            16.7.12  Hypothesis builders   hbntg 23496
            16.7.13  The Predecessor Class   cpred 23501
            16.7.14  (Trans)finite Recursion Theorems   tfisg 23538
            16.7.15  Well-founded induction   tz6.26 23539
            16.7.16  Transitive closure under a relationship   ctrpred 23554
            16.7.17  Founded Induction   frmin 23576
            16.7.18  Ordering Ordinal Sequences   orderseqlem 23586
            16.7.19  Well-founded recursion   wfr3g 23589
            16.7.20  Transfinite recursion via Well-founded recursion   tfrALTlem 23610
            16.7.21  Founded Recursion   frr3g 23614
            16.7.22  Surreal Numbers   csur 23628
            16.7.23  Surreal Numbers: Ordering   axsltsolem1 23655
            16.7.24  Surreal Numbers: Birthday Function   axbday 23662
            16.7.25  Surreal Numbers: Density   axdenselem1 23669
            16.7.26  Surreal Numbers: Full-Eta Property   axfelem1 23680
            16.7.27  Symmetric difference   csymdif 23702
            16.7.28  Quantifier-free definitions   ctxp 23714
            16.7.29  Alternate ordered pairs   caltop 23830
            16.7.30  Tarskian geometry   cee 23856
            16.7.31  Tarski's axioms for geometry   axdimuniq 23881
            16.7.32  Congruence properties   cofs 23945
            16.7.33  Betweenness properties   btwntriv2 23975
            16.7.34  Segment Transportation   ctransport 23992
            16.7.35  Properties relating betweenness and congruence   cifs 23998
            16.7.36  Connectivity of betweenness   btwnconn1lem1 24050
            16.7.37  Segment less than or equal to   csegle 24069
            16.7.38  Outside of relationship   coutsideof 24082
            16.7.39  Lines and Rays   cline2 24097
            16.7.40  Bernoulli polynomials and sums of k-th powers   cbp 24121
            16.7.41  Rank theorems   rankung 24136
            16.7.42  Hereditarily Finite Sets   chf 24142
      16.8  Mathbox for Anthony Hart
            16.8.1  Propositional Calculus   tb-ax1 24157
            16.8.2  Predicate Calculus   quantriv 24179
            16.8.3  Misc. Single Axiom Systems   meran1 24190
            16.8.4  Connective Symmetry   negsym1 24196
      16.9  Mathbox for Chen-Pang He
            16.9.1  Ordinal topology   ontopbas 24207
      16.10  Mathbox for Jeff Hoffman
            16.10.1  Inferences for finite induction on generic function values   fveleq 24230
            16.10.2  gdc.mm   nnssi2 24234
      16.11  Mathbox for Wolf Lammen
      16.12  Mathbox for Frédéric Liné
            16.12.1  Theorems from other workspaces   tpssg 24263
            16.12.2  Propositional and predicate calculus   neleq12d 24264
            16.12.3  Linear temporal logic   wbox 24301
            16.12.4  Operations   ssoprab2g 24363
            16.12.5  General Set Theory   uninqs 24370
            16.12.6  The "maps to" notation   cmpfun 24474
            16.12.7  Cartesian Products   cpro 24482
            16.12.8  Operations on subsets and functions   ccst 24504
            16.12.9  Arithmetic   3timesi 24510
            16.12.10  Lattice (algebraic definition)   clatalg 24513
            16.12.11  Currying and Partial Mappings   ccur1 24526
            16.12.12  Order theory (Extensible Structure Builder)   corhom 24539
            16.12.13  Order theory   cpresetrel 24547
            16.12.14  Finite composites ( i. e. finite sums, products ... )   cprd 24630
            16.12.15  Operation properties   ccm1 24663
            16.12.16  Groups and related structures   ridlideq 24667
            16.12.17  Free structures   csubsmg 24715
            16.12.18  Translations   trdom2 24723
            16.12.19  Fields and Rings   com2i 24748
            16.12.20  Ideals   cidln 24775
            16.12.21  Generic modules and vector spaces (New Structure builder)   cact 24779
            16.12.22  Generic modules and vector spaces   cvec 24781
            16.12.23  Real vector spaces   cvr 24821
            16.12.24  Matrices   cmmat 24825
            16.12.25  Affine spaces   craffsp 24831
            16.12.26  Intervals of reals and extended reals   bsi 24833
            16.12.27  Topology   topnem 24844
            16.12.28  Continuous functions   cnrsfin 24857
            16.12.29  Homeomorphisms   dmhmph 24865
            16.12.30  Initial and final topologies   intopcoaconlem3b 24870
            16.12.31  Filters   efilcp 24884
            16.12.32  Limits   plimfil 24890
            16.12.33  Uniform spaces   cunifsp 24917
            16.12.34  Separated spaces: T0, T1, T2 (Hausdorff) ...   hst1 24919
            16.12.35  Compactness   indcomp 24921
            16.12.36  Connectedness   singempcon 24925
            16.12.37  Topological fields   ctopfld 24929
            16.12.38  Standard topology on RR   intrn 24931
            16.12.39  Standard topology of intervals of RR   stoi 24933
            16.12.40  Cantor's set   cntrset 24934
            16.12.41  Pre-calculus and Cartesian geometry   dmse1 24935
            16.12.42  Extended Real numbers   nolimf 24951
            16.12.43  ( RR ^ N ) and ( CC ^ N )   cplcv 24976
            16.12.44  Calculus   cintvl 25028
            16.12.45  Directed multi graphs   cmgra 25040
            16.12.46  Category and deductive system underlying "structure"   calg 25043
            16.12.47  Deductive systems   cded 25066
            16.12.48  Categories   ccatOLD 25084
            16.12.49  Homsets   chomOLD 25117
            16.12.50  Monomorphisms, Epimorphisms, Isomorphisms   cepiOLD 25135
            16.12.51  Functors   cfuncOLD 25163
            16.12.52  Subcategories   csubcat 25175
            16.12.53  Terminal and initial objects   ciobj 25192
            16.12.54  Sources and sinks   csrce 25197
            16.12.55  Limits and co-limits   clmct 25206
            16.12.56  Product and sum of two objects   cprodo 25209
            16.12.57  Tarski's classes   ctar 25213
            16.12.58  Category Set   ccmrcase 25242
            16.12.59  Grammars, Logics, Machines and Automata   ckln 25312
            16.12.60  Words   cwrd 25313
            16.12.61  Planar geometry   cpoints 25388
      16.13  Mathbox for Jeff Hankins
            16.13.1  Miscellany   a1i13 25532
            16.13.2  Basic topological facts   topbnd 25574
            16.13.3  Topology of the real numbers   reconnOLD 25587
            16.13.4  Refinements   cfne 25591
            16.13.5  Neighborhood bases determine topologies   neibastop1 25640
            16.13.6  Lattice structure of topologies   topmtcl 25644
            16.13.7  Filter bases   fgmin 25651
            16.13.8  Directed sets, nets   tailfval 25653
      16.14  Mathbox for Jeff Madsen
            16.14.1  Logic and set theory   anim12da 25664
            16.14.2  Real and complex numbers; integers   fimaxreOLD 25762
            16.14.3  Sequences and sums   sdclem2 25784
            16.14.4  Topology   unopnOLD 25796
            16.14.5  Metric spaces   metf1o 25801
            16.14.6  Continuous maps and homeomorphisms   constcncf 25810
            16.14.7  Product topologies   txtopiOLD 25818
            16.14.8  Boundedness   ctotbnd 25822
            16.14.9  Isometries   cismty 25854
            16.14.10  Heine-Borel Theorem   heibor1lem 25865
            16.14.11  Banach Fixed Point Theorem   bfplem1 25878
            16.14.12  Euclidean space   crrn 25881
            16.14.13  Intervals (continued)   ismrer1 25894
            16.14.14  Groups and related structures   exidcl 25898
            16.14.15  Rings   rngonegcl 25908
            16.14.16  Ring homomorphisms   crnghom 25923
            16.14.17  Commutative rings   ccring 25952
            16.14.18  Ideals   cidl 25964
            16.14.19  Prime rings and integral domains   cprrng 26003
            16.14.20  Ideal generators   cigen 26016
      16.15  Mathbox for Rodolfo Medina
            16.15.1  Partitions   prtlem60 26035
      16.16  Mathbox for Stefan O'Rear
            16.16.1  Additional elementary logic and set theory   nelss 26083
            16.16.2  Additional theory of functions   fninfp 26086
            16.16.3  Extensions beyond function theory   gsumvsmul 26096
            16.16.4  Additional topology   elrfi 26101
            16.16.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26105
            16.16.6  Algebraic closure systems   cnacs 26109
            16.16.7  Miscellanea 1. Map utilities   constmap 26120
            16.16.8  Miscellanea for polynomials   ofmpteq 26129
            16.16.9  Multivariate polynomials over the integers   cmzpcl 26131
            16.16.10  Miscellanea for Diophantine sets 1   coeq0 26163
            16.16.11  Diophantine sets 1: definitions   cdioph 26166
            16.16.12  Diophantine sets 2 miscellanea   ellz1 26178
            16.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26184
            16.16.14  Diophantine sets 3: construction   diophrex 26187
            16.16.15  Diophantine sets 4 miscellanea   2sbcrex 26196
            16.16.16  Diophantine sets 4: Quantification   rexrabdioph 26207
            16.16.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26214
            16.16.18  Diophantine sets 6 miscellanea   fz1ssnn 26224
            16.16.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26226
            16.16.20  Pigeonhole Principle and cardinality helpers   fphpd 26231
            16.16.21  A non-closed set of reals is infinite   rencldnfilem 26235
            16.16.22  Miscellanea for Lagrange's theorem   icodiamlt 26237
            16.16.23  Lagrange's rational approximation theorem   irrapxlem1 26239
            16.16.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26246
            16.16.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26253
            16.16.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26295
            16.16.27  Logarithm laws generalized to an arbitrary base   reglogcl 26307
            16.16.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26315
            16.16.29  X and Y sequences 1: Definition and recurrence laws   crmx 26317
            16.16.30  Ordering and induction lemmas for the integers   monotuz 26358
            16.16.31  X and Y sequences 2: Order properties   rmxypos 26366
            16.16.32  Congruential equations   congtr 26384
            16.16.33  Alternating congruential equations   acongid 26394
            16.16.34  Additional theorems on integer divisibility   bezoutr 26404
            16.16.35  X and Y sequences 3: Divisibility properties   jm2.18 26413
            16.16.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26430
            16.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26440
            16.16.38  Uncategorized stuff not associated with a major project   setindtr 26449
            16.16.39  More equivalents of the Axiom of Choice   axac10 26458
            16.16.40  Finitely generated left modules   clfig 26497
            16.16.41  Noetherian left modules I   clnm 26505
            16.16.42  Addenda for structure powers   pwssplit0 26519
            16.16.43  Direct sum of left modules   cdsmm 26529
            16.16.44  Free modules   cfrlm 26544
            16.16.45  Every set admits a group structure iff choice   unxpwdom3 26588
            16.16.46  Independent sets and families   clindf 26606
            16.16.47  Characterization of free modules   lmimlbs 26638
            16.16.48  Noetherian rings and left modules II   clnr 26645
            16.16.49  Hilbert's Basis Theorem   cldgis 26657
            16.16.50  Additional material on polynomials [DEPRECATED]   cmnc 26667
            16.16.51  Degree and minimal polynomial of algebraic numbers   cdgraa 26677
            16.16.52  Algebraic integers I   citgo 26694
            16.16.53  Finite cardinality [SO]   en1uniel 26712
            16.16.54  Words in monoids and ordered group sum   issubmd 26715
            16.16.55  Transpositions in the symmetric group   cpmtr 26716
            16.16.56  The sign of a permutation   cpsgn 26746
            16.16.57  The matrix algebra   cmmul 26771
            16.16.58  The determinant   cmdat 26815
            16.16.59  Endomorphism algebra   cmend 26821
            16.16.60  Subfields   csdrg 26835
            16.16.61  Cyclic groups and order   idomrootle 26843
            16.16.62  Cyclotomic polynomials   ccytp 26853
            16.16.63  Miscellaneous topology   fgraphopab 26861
      16.17  Mathbox for Steve Rodriguez
            16.17.1  Miscellanea   iso0 26868
            16.17.2  Function operations   caofcan 26872
            16.17.3  Calculus   lhe4.4ex1a 26878
      16.18  Mathbox for Andrew Salmon
            16.18.1  Principia Mathematica * 10   pm10.12 26885
            16.18.2  Principia Mathematica * 11   2alanimi 26899
            16.18.3  Predicate Calculus   sbeqal1 26929
            16.18.4  Principia Mathematica * 13 and * 14   pm13.13a 26940
            16.18.5  Set Theory   elnev 26971
            16.18.6  Arithmetic   addcomgi 26994
            16.18.7  Geometry   cplusr 26995
      16.19  Mathbox for Glauco Siliprandi
            16.19.1  Miscellanea   ssrexf 27017
            16.19.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27043
            16.19.3  Stone Weierstrass theorem - real version   stoweidlem1 27050
      16.20  Mathbox for Jarvin Udandy
      16.21  Mathbox for David A. Wheeler
            16.21.1  Natural deduction   19.8ad 27199
            16.21.2  Greater than, greater than or equal to.   cge-real 27202
            16.21.3  Hyperbolic trig functions   csinh 27212
            16.21.4  Reciprocal trig functions (sec, csc, cot)   csec 27223
            16.21.5  Identities for "if"   ifnmfalse 27245
            16.21.6  Not-member-of   AnelBC 27246
            16.21.7  Decimal point   cdp2 27247
            16.21.8  Signum (sgn or sign) function   csgn 27255
            16.21.9  Ceiling function   ccei 27265
            16.21.10  Logarithm laws generalized to an arbitrary base   clogb 27269
            16.21.11  Miscellaneous   2m1e1 27274
      16.22  Mathbox for Alan Sare
            16.22.1  Conventional Metamath proofs, some derived from VD proofs   iidn3 27278
            16.22.2  What is Virtual Deduction?   wvd1 27353
            16.22.3  Virtual Deduction Theorems   df-vd1 27354
            16.22.4  Theorems proved using virtual deduction   trsspwALT 27605
            16.22.5  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 27635
            16.22.6  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 27702
            16.22.7  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 27706
            16.22.8  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 27713
            16.22.9  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 27716
      16.23  Mathbox for Jonathan Ben-Naim
            16.23.1  First order logic and set theory   bnj170 27735
            16.23.2  Well founded induction and recursion   bnj110 27902
            16.23.3  The existence of a minimal element in certain classes   bnj69 28052
            16.23.4  Well-founded induction   bnj1204 28054
            16.23.5  Well-founded recursion, part 1 of 3   bnj60 28104
            16.23.6  Well-founded recursion, part 2 of 3   bnj1500 28110
            16.23.7  Well-founded recursion, part 3 of 3   bnj1522 28114
      16.24  Mathbox for Norm Megill
            16.24.1  Study of ax-6, ax-7, ax-11, ax-12   equidK 28115
            16.24.2  Derive ax-12o from ax-12   ax12vX 28161
            16.24.3  Derive ax-10   ax10lem16X 28215
            16.24.4  Derive ax-9 from the weaker version ax-9v   ax9X 28232
            16.24.5  Obsolete experiments to study ax-12o   ax12-2 28233
            16.24.6  Miscellanea   cnaddcom 28291
            16.24.7  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 28294
            16.24.8  Functionals and kernels of a left vector space (or module)   clfn 28377
            16.24.9  Opposite rings and dual vector spaces   cld 28443
            16.24.10  Ortholattices and orthomodular lattices   cops 28492
            16.24.11  Atomic lattices with covering property   ccvr 28582
            16.24.12  Hilbert lattices   chlt 28670
            16.24.13  Projective geometries based on Hilbert lattices   clln 28810
            16.24.14  Construction of a vector space from a Hilbert lattice   cdlema1N 29110
            16.24.15  Construction of involution and inner product from a Hilbert lattice   clpoN 30800

    < Wrap  Next >

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 31201-31284
  Copyright terms: Public domain < Wrap  Next >