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 Mario Carneiro
      16.4  Mathbox for Paul Chapman
      16.5  Mathbox for Drahflow
      16.6  Mathbox for Scott Fenton
      16.7  Mathbox for Anthony Hart
      16.8  Mathbox for Chen-Pang He
      16.9  Mathbox for Jeff Hoffman
      16.10  Mathbox for Wolf Lammen
      16.11  Mathbox for Frédéric Liné
      16.12  Mathbox for Jeff Hankins
      16.13  Mathbox for Jeff Madsen
      16.14  Mathbox for Rodolfo Medina
      16.15  Mathbox for Stefan O'Rear
      16.16  Mathbox for Steve Rodriguez
      16.17  Mathbox for Andrew Salmon
      16.18  Mathbox for Jarvin Udandy
      16.19  Mathbox for David A. Wheeler
      16.20  Mathbox for Alan Sare
      16.21  Mathbox for Jonathan Ben-Naim
      16.22  Mathbox for Norm Megill

Detailed Table of Contents
PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
            1.1.1  Inferences for assisting proof development   dummylink 1
      1.2  Conventions
      1.3  Propositional calculus
            1.3.1  Recursively define primitive wffs for propositional calculus   wn 5
            1.3.2  The axioms of propositional calculus   ax-1 7
            1.3.3  Logical implication   mp2b 11
            1.3.4  Logical negation   con4d 99
            1.3.5  Logical equivalence   wb 178
            1.3.6  Logical disjunction and conjunction   wo 359
            1.3.7  Miscellaneous theorems of propositional calculus   pm5.21nd 873
            1.3.8  Abbreviated conjunction and disjunction of three wff's   w3o 938
            1.3.9  Logical 'nand' (Sheffer stroke)   wnan 1292
            1.3.10  Logical 'xor'   wxo 1300
            1.3.11  True and false constants   wtru 1312
            1.3.12  Truth tables   truantru 1332
            1.3.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1358
            1.3.14  Half-adders and full adders in propositional calculus   whad 1374
      1.4  Other axiomatizations of classical propositional calculus
            1.4.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1400
            1.4.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1418
            1.4.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1429
            1.4.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1435
            1.4.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1454
            1.4.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1458
            1.4.7  Deriving the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1473
            1.4.8  Deriving the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1496
            1.4.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1509
            1.4.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1528
      1.5  Predicate calculus mostly without distinct variables
            1.5.1  "Pure" (equality-free) predicate calculus axioms ax-5, ax-6, ax-7, ax-gen   wal 1532
            1.5.2  Introduce equality axioms ax-8, ax-11, ax-13, and ax-14   cv 1618
            1.5.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1628
            1.5.4  Introduce equality axioms ax-9v and ax-12   ax-9v 1632
            1.5.5  Derive ax-12o from ax-12   ax12o10lem1 1635
            1.5.6  Derive ax-10   ax10lem16 1665
            1.5.7  Derive ax-9 from the weaker version ax-9v   ax9 1683
            1.5.8  Introduce Axiom of Existence ax-9   ax-9 1684
            1.5.9  Derive ax-4, ax-5o, and ax-6o   ax4 1691
            1.5.10  "Pure" predicate calculus including ax-4, without distinct variables   a4i 1699
            1.5.11  Equality theorems without distinct variables   ax9o 1814
            1.5.12  Axioms ax-10 and ax-11   ax10o 1834
            1.5.13  Substitution (without distinct variables)   wsb 1882
            1.5.14  Theorems using axiom ax-11   equs5a 1911
      1.6  Predicate calculus with distinct variables
            1.6.1  Derive the axiom of distinct variables ax-16   a4imv 1922
            1.6.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1939
            1.6.3  Theorems without distinct variables that use axiom ax-11o   ax11b 1942
            1.6.4  Predicate calculus with distinct variables (cont.)   ax11v 1990
            1.6.5  More substitution theorems   equsb3lem 2061
            1.6.6  Existential uniqueness   weu 2114
      1.7  Other axiomatizations related to classical predicate calculus
            1.7.1  Predicate calculus with all distinct variables   ax-7d 2204
            1.7.2  Aristotelian logic: Assertic syllogisms   barbara 2210
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
            2.1.1  Introduce the Axiom of Extensionality   ax-ext 2234
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2239
            2.1.3  Class form not-free predicate   wnfc 2372
            2.1.4  Negated equality and membership   wne 2412
            2.1.5  Restricted quantification   wral 2507
            2.1.6  The universal class   cvv 2725
            2.1.7  Conditional equality (experimental)   wcdeq 2902
            2.1.8  Russell's Paradox   ru 2918
            2.1.9  Proper substitution of classes for sets   wsbc 2919
            2.1.10  Proper substitution of classes for sets into classes   csb 3006
            2.1.11  Define basic set operations and relations   cdif 3072
            2.1.12  Subclasses and subsets   df-ss 3086
            2.1.13  The difference, union, and intersection of two classes   difeq1 3201
            2.1.14  The empty set   c0 3359
            2.1.15  "Weak deduction theorem" for set theory   cif 3467
            2.1.16  Power classes   cpw 3527
            2.1.17  Unordered and ordered pairs   csn 3541
            2.1.18  The union of a class   cuni 3724
            2.1.19  The intersection of a class   cint 3757
            2.1.20  Indexed union and intersection   ciun 3800
            2.1.21  Disjointness   wdisj 3888
            2.1.22  Binary relations   wbr 3917
            2.1.23  Ordered-pair class abstractions (class builders)   copab 3970
            2.1.24  Transitive classes   wtr 4007
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4025
            2.2.2  Derive the Axiom of Separation   axsep 4034
            2.2.3  Derive the Null Set Axiom   zfnuleu 4040
            2.2.4  Theorems requiring subset and intersection existence   nalset 4045
            2.2.5  Theorems requiring empty set existence   class2set 4069
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4079
            2.3.2  Derive the Axiom of Pairing   zfpair 4103
            2.3.3  Ordered pair theorem   opnz 4132
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4161
            2.3.5  Power class of union and intersection   pwin 4187
            2.3.6  Epsilon and identity relations   cep 4193
            2.3.7  Partial and complete ordering   wpo 4202
            2.3.8  Founded and well-ordering relations   wfr 4239
            2.3.9  Ordinals   word 4281
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4400
            2.4.2  Ordinals (continued)   ordon 4462
            2.4.3  Transfinite induction   tfi 4532
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4544
            2.4.5  Peano's postulates   peano1 4563
            2.4.6  Finite induction (for finite ordinals)   find 4569
            2.4.7  Functions and relations   cxp 4575
            2.4.8  Operations   co 5707
            2.4.9  "Maps to" notation   elmpt2cl 5910
            2.4.10  Function operation   cof 5925
            2.4.11  First and second members of an ordered pair   c1st 5969
            2.4.12  Function transposition   ctpos 6082
            2.4.13  Curry and uncurry   ccur 6121
            2.4.14  Proper subset relation   crpss 6125
            2.4.15  Definite description binder (inverted iota)   cio 6138
            2.4.16  Cantor's Theorem   canth 6175
            2.4.17  Undefined values and restricted iota (description binder)   cund 6177
            2.4.18  Functions on ordinals; strictly monotone ordinal functions   iunon 6238
            2.4.19  "Strong" transfinite recursion   crecs 6270
            2.4.20  Recursive definition generator   crdg 6305
            2.4.21  Finite recursion   frfnom 6330
            2.4.22  Abian's "most fundamental" fixed point theorem   abianfplem 6353
            2.4.23  Ordinal arithmetic   c1o 6355
            2.4.24  Natural number arithmetic   nna0 6485
            2.4.25  Equivalence relations and classes   wer 6540
            2.4.26  The mapping operation   cmap 6655
            2.4.27  Infinite Cartesian products   cixp 6700
            2.4.28  Equinumerosity   cen 6743
            2.4.29  Schroeder-Bernstein Theorem   sbthlem1 6853
            2.4.30  Equinumerosity (cont.)   xpf1o 6905
            2.4.31  Pigeonhole Principle   phplem1 6922
            2.4.32  Finite sets   onomeneq 6932
            2.4.33  Finite intersections   cfi 7045
            2.4.34  Hall's marriage theorem   marypha1lem 7067
            2.4.35  Supremum   csup 7074
            2.4.36  Ordinal isomorphism, Hartog's theorem   coi 7105
            2.4.37  Hartogs function, order types, weak dominance   char 7151
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7187
            2.5.2  Axiom of Infinity equivalents   inf0 7203
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7220
            2.6.2  Existence of omega (the set of natural numbers)   omex 7225
            2.6.3  Cantor normal form   ccnf 7243
            2.6.4  Transitive closure   trcl 7291
            2.6.5  Rank   cr1 7315
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7436
            2.6.7  Cardinal numbers   ccrd 7449
            2.6.8  Axiom of Choice equivalents   wac 7623
            2.6.9  Cardinal number arithmetic   ccda 7674
            2.6.10  The Ackermann bijection   ackbij2lem1 7726
            2.6.11  Cofinality (without Axiom of Choice)   cflem 7753
            2.6.12  Eight inequivalent definitions of finite set   sornom 7784
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 7923
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 7966
            3.2.2  AC equivalents: well ordering, Zorn's lemma   numthcor 8002
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8049
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8073
            3.2.5  Cofinality using Axiom of Choice   alephreg 8081
      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 8181
            4.1.2  Weak universes   cwun 8199
            4.1.3  Tarski's classes   ctsk 8247
            4.1.4  Grothendieck's universes   cgru 8289
      4.2  ZFC Set Theory plus the Tarksi-Grothendieck Axiom
            4.2.1  Introduce the Tarksi-Grothendieck Axiom   ax-groth 8322
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8325
            4.2.3  Tarski map function   ctskm 8336
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 8343
            5.1.2  Final derivation of real and complex number postulates   axaddf 8644
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8670
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 8695
            5.2.2  Infinity and the extended real number system   cpnf 8741
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 8770
            5.2.4  Ordering on reals   lttr 8775
            5.2.5  Initial properties of the complex numbers   mul12 8854
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 8897
            5.3.2  Subtraction   cmin 8909
            5.3.3  Multiplication   muladd 9073
            5.3.4  Ordering on reals (cont.)   gt0ne0 9100
            5.3.5  Reciprocals   ixi 9257
            5.3.6  Division   cdiv 9279
            5.3.7  Ordering on reals (cont.)   elimgt0 9440
            5.3.8  Completeness Axiom and Suprema   fimaxre 9549
            5.3.9  Imaginary and complex number properties   inelr 9584
            5.3.10  Function operation analogue theorems   ofsubeq0 9591
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9594
            5.4.2  Principle of mathematical induction   nnind 9612
            5.4.3  Decimal representation of numbers   c2 9643
            5.4.4  Some properties of specific numbers   0p1e1 9687
            5.4.5  The Archimedean property   nnunb 9807
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 9811
            5.4.7  Integers (as a subset of complex numbers)   cz 9870
            5.4.8  Decimal arithmetic   cdc 9970
            5.4.9  Upper partititions of integers   cuz 10076
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10157
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10162
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10188
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10200
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10295
            5.5.3  Supremum on the extended reals   xrsupexmnf 10468
            5.5.4  Real number intervals   cioo 10501
            5.5.5  Finite intervals of integers   cfz 10625
            5.5.6  Half-open integer ranges   cfzo 10713
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 10767
            5.6.2  The modulo (remainder) operation   cmo 10816
            5.6.3  The infinite sequence builder "seq"   om2uz0i 10853
            5.6.4  Integer powers   cexp 10947
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11124
            5.6.6  Factorial function   cfa 11130
            5.6.7  The binomial coefficient operation   cbc 11157
            5.6.8  The ` # ` (finite set size) function   chash 11179
            5.6.9  Words over a set   cword 11244
            5.6.10  Longer string literals   cs2 11332
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11402
            5.7.2  Real and imaginary parts; conjugate   ccj 11422
            5.7.3  Square root; absolute value   csqr 11559
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 11783
            5.8.2  Limits   cli 11797
            5.8.3  Finite and infinite sums   csu 11997
            5.8.4  The binomial theorem   binomlem 12126
            5.8.5  Infinite sums (cont.)   isumshft 12134
            5.8.6  Miscellaneous converging and diverging sequences   divrcnv 12147
            5.8.7  Arithmetic series   arisum 12154
            5.8.8  Geometric series   expcnv 12158
            5.8.9  Ratio test for infinite series convergence   cvgrat 12175
            5.8.10  Mertens' theorem   mertenslem1 12176
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12179
            5.9.2  _e is irrational   eirrlem 12318
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12323
            5.10.2  The reals are uncountable   rpnnen2lem1 12329
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12362
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12365
            6.1.3  The divides relation   cdivides 12367
            6.1.4  The division algorithm   divalglem0 12428
            6.1.5  Bit sequences   cbits 12446
            6.1.6  The greatest common divisor operator   cgcd 12521
            6.1.7  Bézout's identity   bezoutlem1 12553
            6.1.8  Algorithms   nn0seqcvgd 12576
            6.1.9  Euclid's Algorithm   eucalgval2 12587
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 12594
            6.2.2  Properties of the canonical representation of a rational   cnumer 12640
            6.2.3  Euler's theorem   codz 12667
            6.2.4  Pythagorean Triples   coprimeprodsq 12698
            6.2.5  The prime count function   cpc 12725
            6.2.6  Pocklington's theorem   prmpwdvds 12787
            6.2.7  Infinite primes theorem   unbenlem 12791
            6.2.8  Sum of prime reciprocals   prmreclem1 12799
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 12806
            6.2.10  Lagrange's four-square theorem   cgz 12812
            6.2.11  Van der Waerden's theorem   cvdwa 12848
            6.2.12  Ramsey's theorem   cram 12882
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 12914
            6.2.14  Specific prime numbers   4nprm 12942
            6.2.15  Very large primes   1259lem1 12965
PART 7  EXTENSIBLE STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 12980
            7.1.2  Slot definitions   cplusg 13044
            7.1.3  Definition of the structure product   crest 13161
            7.1.4  Definition of the structure quotient   cordt 13234
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13342
            7.2.2  Algebraic closure systems   isacs 13360
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13372
            8.1.2  Opposite category   coppc 13420
            8.1.3  Monomorphisms and epimorphisms   cmon 13437
            8.1.4  Sections, inverses, isomorphisms   csect 13453
            8.1.5  Subcategories   cssc 13490
            8.1.6  Functors   cfunc 13534
            8.1.7  Full & faithful functors   cful 13582
            8.1.8  Natural transformations and the functor category   cnat 13621
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 13691
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 13713
            8.3.2  The category of categories   ccatc 13732
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 13748
            8.4.2  Functor evaluation   cevlf 13789
            8.4.3  Hom functor   chof 13828
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 13880
            9.2.2  Lattices   clat 13957
            9.2.3  The dual of an ordered set   codu 14038
            9.2.4  Subset order structures   cipo 14060
            9.2.5  Distributive lattices   latmass 14088
            9.2.6  Posets and lattices as relations   cps 14098
            9.2.7  Directed sets, nets   cdir 14147
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14158
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14210
            10.1.3  Ordered group sum operation   gsumvallem1 14245
            10.1.4  Free monoids   cfrmd 14266
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14286
            10.2.2  Subgroups and Quotient groups   csubg 14412
            10.2.3  Elementary theory of group homomorphisms   cghm 14477
            10.2.4  Isomorphisms of groups   cgim 14518
            10.2.5  Group actions   cga 14540
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 14566
            10.2.7  Centralizers and centers   ccntz 14588
            10.2.8  The opposite group   coppg 14615
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 14637
            10.2.10  Direct products   clsm 14742
            10.2.11  Free groups   cefg 14812
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 14886
            10.3.2  Cyclic groups   ccyg 14961
            10.3.3  Group sum operation   gsumval3a 14986
            10.3.4  Internal direct products   cdprd 15028
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15097
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15122
            10.4.2  Definition and basic properties   crg 15134
            10.4.3  Opposite ring   coppr 15201
            10.4.4  Divisibility   cdsr 15217
            10.4.5  Ring homomorphisms   crh 15291
      10.5  Division rings and Fields
            10.5.1  Definition and basic properties   cdr 15309
            10.5.2  Subrings of a ring   csubrg 15338
            10.5.3  Absolute value (abstract algebra)   cabv 15378
            10.5.4  Star rings   cstf 15405
      10.6  Left Modules
            10.6.1  Definition and basic properties   clmod 15424
            10.6.2  Subspaces and spans in a left module   clss 15486
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 15573
            10.6.4  Subspace sum; bases for a left module   clbs 15624
      10.7  Vector Spaces
            10.7.1  Definition and basic properties   clvec 15652
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 15715
            10.8.2  Two-sided ideals and quotient rings   c2idl 15777
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 15787
            10.8.4  Nonzero rings   cnzr 15803
            10.8.5  Left regular elements. More kinds of ring   crlreg 15814
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 15844
      10.10  Abstract Multivariate Polynomials
            10.10.1  Definition and basic properties   cmps 15881
            10.10.2  Polynomial evaluation   evlslem4 16039
            10.10.3  Univariate Polynomials   cps1 16044
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16163
            10.11.2  Algebraic constructions based on the complexes   czrh 16245
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16322
            10.12.2  Orthocomplements and closed subspaces   cocv 16354
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16394
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16425
            11.1.2  TopBases for topologies   isbasisg 16479
            11.1.3  Examples of topologies   distop 16527
            11.1.4  Closure and interior   ccld 16547
            11.1.5  Neighborhoods   cnei 16628
            11.1.6  Limit points and perfect sets   clp 16660
            11.1.7  Subspace topologies   restrcl 16682
            11.1.8  Order topology   ordtbaslem 16712
            11.1.9  Limits and Continuity in topological spaces   ccn 16748
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 16828
            11.1.11  Compactness   ccmp 16907
            11.1.12  Connectedness   ccon 16931
            11.1.13  First- and second-countability   c1stc 16957
            11.1.14  Local topological properties   clly 16984
            11.1.15  Compactly generated spaces   ckgen 17022
            11.1.16  Product topologies   ctx 17049
            11.1.17  Continuous function-builders   cnmptid 17149
            11.1.18  Quotient maps and quotient topology   ckq 17178
            11.1.19  Homeomorphisms   chmeo 17238
      11.2  Filters and filter bases
            11.2.1  Filter Bases   cfbas 17312
            11.2.2  Filters   cfil 17334
            11.2.3  Ultrafilters   cufil 17388
            11.2.4  Filter limits   cfm 17422
            11.2.5  Topological groups   ctmd 17547
            11.2.6  Infinite group sum on topological groups   ctsu 17602
            11.2.7  Topological rings, fields, vector spaces   ctrg 17632
      11.3  Metric spaces
            11.3.1  Basic metric space properties   cxme 17676
            11.3.2  Metric space balls   blfval 17741
            11.3.3  Open sets of a metric space   mopnval 17778
            11.3.4  Continuity in metric spaces   metcnp3 17880
            11.3.5  Examples of metric spaces   dscmet 17889
            11.3.6  Normed algebraic structures   cnm 17893
            11.3.7  Normed space homomorphisms (bounded linear operators)   cnmo 18008
            11.3.8  Topology on the Reals   qtopbaslem 18061
            11.3.9  Topological definitions using the reals   cii 18173
            11.3.10  Path homotopy   chtpy 18259
            11.3.11  The fundamental group   cpco 18292
            11.3.12  Complex left modules   cclm 18354
            11.3.13  Complex pre-Hilbert space   ccph 18396
            11.3.14  Convergence and completeness   ccfil 18472
            11.3.15  Baire's Category Theorem   bcthlem1 18540
            11.3.16  Banach spaces and complex Hilbert spaces   ccms 18548
            11.3.17  Minimizing Vector Theorem   minveclem1 18582
            11.3.18  Projection theorem   pjthlem1 18595
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 18602
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 18616
            12.2.2  Lebesgue integration   cmbf 18763
      12.3  Derivatives
            12.3.1  Real and Complex Differentiation   climc 19006
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19191
            13.1.2  Polynomial degrees   cmdg 19233
            13.1.3  The division algorithm for univariate polynomials   cmn1 19305
            13.1.4  Elementary properties of complex polynomials   cply 19360
            13.1.5  The Division algorithm for polynomials   cquot 19464
            13.1.6  Algebraic numbers   caa 19488
            13.1.7  Liouville's approximation theorem   aalioulem1 19506
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 19526
            13.2.2  Uniform convergence   culm 19549
            13.2.3  Power series   pserval 19580
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 19613
            13.3.2  Properties of pi = 3.14159...   pilem1 19621
            13.3.3  Mapping of the exponential function   efgh 19692
            13.3.4  The natural logarithm on complex numbers   clog 19701
            13.3.5  Solutions of quardatic, cubic, and quartic equations   quad2 19901
            13.3.6  Inverse trigonometric functions   casin 19924
            13.3.7  The Birthday Problem   log2ublem1 20008
            13.3.8  Areas in R^2   carea 20016
            13.3.9  More miscellaneous converging sequences   rlimcnp 20026
            13.3.10  Inequality of arithmetic and geometric means   cvxcl 20045
            13.3.11  Euler-Mascheroni constant   cem 20052
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20072
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20076
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20084
            13.4.4  Number-theoretical functions   ccht 20094
            13.4.5  Perfect Number Theorem   mersenne 20232
            13.4.6  Characters of Z/nZ   cdchr 20237
            13.4.7  Bertrand's postulate   bcctr 20280
            13.4.8  Legendre symbol   clgs 20299
            13.4.9  Quadratic Reciprocity   lgseisenlem1 20354
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 20368
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 20384
            13.4.12  The Prime Number Theorem   mudivsum 20445
            13.4.13  Ostrowski's theorem   abvcxp 20530
PART 14  MISCELLANEA
      14.1  Definitional Examples
      14.2  Natural deduction examples
      14.3  Humor
            14.3.1  April Fool's theorem   avril1 20595
      14.4  (Future - to be reviewed and classified)
            14.4.1  Planar incidence geometry   cplig 20601
            14.4.2  Algebra preliminaries   crpm 20606
            14.4.3  Transitive closure   ctcl 20608
PART 15  DEPRECATED SECTIONS
      15.1  Additional material on Group theory
            15.1.1  Definitions and basic properties for groups   cgr 20612
            15.1.2  Definition and basic properties of Abelian groups   cablo 20707
            15.1.3  Subgroups   csubgo 20727
            15.1.4  Operation properties   cass 20738
            15.1.5  Group-like structures   cmagm 20744
            15.1.6  Examples of Abelian groups   ablosn 20773
            15.1.7  Group homomorphism and isomorphism   cghom 20783
      15.2  Additional material on Rings and Fields
            15.2.1  Definition and basic properties   crngo 20801
            15.2.2  Examples of rings   cnrngo 20829
            15.2.3  Division Rings   cdrng 20831
            15.2.4  Star Fields   csfld 20834
            15.2.5  Fields and Rings   ccm2 20836
      15.3  Complex vector spaces
            15.3.1  Definition and basic properties   cvc 20860
            15.3.2  Examples of complex vector spaces   cncvc 20898
      15.4  Normed complex vector spaces
            15.4.1  Definition and basic properties   cnv 20899
            15.4.2  Examples of normed complex vector spaces   cnnv 21004
            15.4.3  Induced metric of a normed complex vector space   imsval 21013
            15.4.4  Inner product   cdip 21032
            15.4.5  Subspaces   css 21056
      15.5  Operators on complex vector spaces
            15.5.1  Definitions and basic properties   clno 21077
      15.6  Inner product (pre-Hilbert) spaces
            15.6.1  Definition and basic properties   ccphlo 21149
            15.6.2  Examples of pre-Hilbert spaces   cncph 21156
            15.6.3  Properties of pre-Hilbert spaces   isph 21159
      15.7  Complex Banach spaces
            15.7.1  Definition and basic properties   ccbn 21200
            15.7.2  Examples of complex Banach spaces   cnbn 21207
            15.7.3  Uniform Boundedness Theorem   ubthlem1 21208
            15.7.4  Minimizing Vector Theorem   minvecolem1 21212
      15.8  Complex Hilbert spaces
            15.8.1  Definition and basic properties   chlo 21223
            15.8.2  Standard axioms for a complex Hilbert space   hlex 21236
            15.8.3  Examples of complex Hilbert spaces   cnchl 21254
            15.8.4  Subspaces   ssphl 21255
            15.8.5  Hellinger-Toeplitz Theorem   htthlem 21256
      15.9  Hilbert Space Explorer
            15.9.1  Basic Hilbert space definitions   chil 21258
            15.9.2  Preliminary ZFC lemmas   df-hnorm 21307
            15.9.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 21320
            15.9.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 21338
            15.9.5  Vector operations   hvmulex 21350
            15.9.6  Inner product postulates for a Hilbert space   ax-hfi 21417
            15.9.7  Inner product   his5 21424
            15.9.8  Norms   dfhnorm2 21460
            15.9.9  Relate Hilbert space to normed complex vector spaces   hilablo 21498
            15.9.10  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 21517
            15.9.11  Cauchy sequences and limits   hcau 21522
            15.9.12  Derivation of the completeness axiom from ZF set theory   hilmet 21532
            15.9.13  Completeness postulate for a Hilbert space   ax-hcompl 21540
            15.9.14  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 21541
            15.9.15  Subspaces   df-sh 21545
            15.9.16  Closed subspaces   df-ch 21560
            15.9.17  Orthocomplements   df-oc 21590
            15.9.18  Subspace sum, span, lattice join, lattice supremum   df-shs 21646
            15.9.19  Projection theorem   pjhthlem1 21729
            15.9.20  Projectors   df-pjh 21733
            15.9.21  Orthomodular law   omlsilem 21740
            15.9.22  Projectors (cont.)   pjhtheu2 21754
            15.9.23  Hilbert lattice operations   sh0le 21778
            15.9.24  Span (cont.) and one-dimensional subspaces   spansn0 21879
            15.9.25  Operator sum, difference, and scalar multiplication   df-hosum 21921
            15.9.26  Commutes relation for Hilbert lattice elements   df-cm 21939
            15.9.27  Foulis-Holland theorem   fh1 21974
            15.9.28  Quantum Logic Explorer axioms   qlax1i 21983
            15.9.29  Orthogonal subspaces   chscllem1 21993
            15.9.30  Orthoarguesian laws 5OA and 3OA   5oalem1 22010
            15.9.31  Projectors (cont.)   pjorthi 22025
            15.9.32  Mayet's equation E_3   mayete3i 22084
            15.9.33  Zero and identity operators   df-h0op 22087
            15.9.34  Operations on Hilbert space operators   hoaddcl 22097
            15.9.35  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 22178
            15.9.36  Linear and continuous functionals and norms   df-nmfn 22184
            15.9.37  Adjoint   df-adjh 22188
            15.9.38  Dirac bra-ket notation   df-bra 22189
            15.9.39  Positive operators   df-leop 22191
            15.9.40  Eigenvectors, eigenvalues, spectrum   df-eigvec 22192
            15.9.41  Theorems about operators and functionals   nmopval 22195
            15.9.42  Riesz lemma   riesz3i 22401
            15.9.43  Adjoints (cont.)   cnlnadjlem1 22406
            15.9.44  Quantum computation error bound theorem   unierri 22443
            15.9.45  Dirac bra-ket notation (cont.)   branmfn 22444
            15.9.46  Positive operators (cont.)   leopg 22461
            15.9.47  Projectors as operators   pjhmopi 22485
            15.9.48  States on a Hilbert lattice   df-st 22550
            15.9.49  Godowski's equation   golem1 22610
            15.9.50  Covers relation; modular pairs   df-cv 22618
            15.9.51  Atoms   df-at 22677
            15.9.52  Superposition principle   superpos 22693
            15.9.53  Atoms, exchange and covering properties, atomicity   chcv1 22694
            15.9.54  Irreducibility   chirredlem1 22729
            15.9.55  Atoms (cont.)   atcvat3i 22735
            15.9.56  Modular symmetry   mdsymlem1 22742
PART 16  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      16.1  Mathboxes for user contributions
            16.1.1  Mathbox guidelines   mathbox 22781
      16.2  Mathbox for Stefan Allan
      16.3  Mathbox for Mario Carneiro
            16.3.1  Miscellaneous stuff   quartfull 22786
            16.3.2  Zeta function   czeta 22787
            16.3.3  Gamma function   clgam 22790
            16.3.4  Derangements and the Subfactorial   deranglem 22797
            16.3.5  The Erdős-Szekeres theorem   erdszelem1 22822
            16.3.6  The Kuratowski closure-complement theorem   kur14lem1 22837
            16.3.7  Retracts and sections   cretr 22848
            16.3.8  Path-connected and simply connected spaces   cpcon 22850
            16.3.9  Covering maps   ccvm 22886
            16.3.10  Undirected multigraphs   cumg 22960
            16.3.11  Normal numbers   snmlff 23012
            16.3.12  Godel-sets of formulas   cgoe 23016
            16.3.13  Models of ZF   cgze 23044
            16.3.14  Splitting fields   citr 23058
            16.3.15  p-adic number fields   czr 23074
      16.4  Mathbox for Paul Chapman
            16.4.1  Group homomorphism and isomorphism   ghomgrpilem1 23092
            16.4.2  Real and complex numbers (cont.)   climuzcnv 23104
            16.4.3  Miscellaneous theorems   elfzm12 23108
      16.5  Mathbox for Drahflow
      16.6  Mathbox for Scott Fenton
            16.6.1  ZFC Axioms in primitive form   axextprim 23147
            16.6.2  Untangled classes   untelirr 23154
            16.6.3  Extra propositional calculus theorems   3orel1 23161
            16.6.4  Misc. Useful Theorems   nepss 23172
            16.6.5  Properties of reals and complexes   sqdivzi 23178
            16.6.6  Greatest common divisor and divisibility   pdivsq 23201
            16.6.7  Properties of relationships   brtp 23205
            16.6.8  Properties of functions and mappings   funpsstri 23218
            16.6.9  Epsilon induction   setinds 23231
            16.6.10  Ordinal numbers   elpotr 23234
            16.6.11  Defined equality axioms   axextdfeq 23251
            16.6.12  Hypothesis builders   hbntg 23259
            16.6.13  The Predecessor Class   cpred 23264
            16.6.14  (Trans)finite Recursion Theorems   tfisg 23301
            16.6.15  Well-founded induction   tz6.26 23302
            16.6.16  Transitive closure under a relationship   ctrpred 23317
            16.6.17  Founded Induction   frmin 23339
            16.6.18  Ordering Ordinal Sequences   orderseqlem 23349
            16.6.19  Well-founded recursion   wfr3g 23352
            16.6.20  Transfinite recursion via Well-founded recursion   tfrALTlem 23373
            16.6.21  Founded Recursion   frr3g 23377
            16.6.22  Surreal Numbers   csur 23391
            16.6.23  Surreal Numbers: Ordering   axsltsolem1 23418
            16.6.24  Surreal Numbers: Birthday Function   axbday 23425
            16.6.25  Surreal Numbers: Density   axdenselem1 23432
            16.6.26  Surreal Numbers: Full-Eta Property   axfelem1 23443
            16.6.27  Symmetric difference   csymdif 23465
            16.6.28  Quantifier-free definitions   ctxp 23477
            16.6.29  Alternate ordered pairs   caltop 23593
            16.6.30  Tarskian geometry   cee 23619
            16.6.31  Tarski's axioms for geometry   axdimuniq 23644
            16.6.32  Congruence properties   cofs 23708
            16.6.33  Betweenness properties   btwntriv2 23738
            16.6.34  Segment Transportation   ctransport 23755
            16.6.35  Properties relating betweenness and congruence   cifs 23761
            16.6.36  Connectivity of betweenness   btwnconn1lem1 23813
            16.6.37  Segment less than or equal to   csegle 23832
            16.6.38  Outside of relationship   coutsideof 23845
            16.6.39  Lines and Rays   cline2 23860
            16.6.40  Bernoulli polynomials and sums of k-th powers   cbp 23884
            16.6.41  Rank theorems   rankung 23899
            16.6.42  Hereditarily Finite Sets   chf 23905
      16.7  Mathbox for Anthony Hart
            16.7.1  Propositional Calculus   tb-ax1 23920
            16.7.2  Predicate Calculus   quantriv 23942
            16.7.3  Misc. Single Axiom Systems   meran1 23953
            16.7.4  Connective Symmetry   negsym1 23959
      16.8  Mathbox for Chen-Pang He
            16.8.1  Ordinal topology   ontopbas 23970
      16.9  Mathbox for Jeff Hoffman
            16.9.1  Inferences for finite induction on generic function values   fveleq 23993
            16.9.2  gdc.mm   nnssi2 23997
      16.10  Mathbox for Wolf Lammen
      16.11  Mathbox for Frédéric Liné
            16.11.1  Theorems from other workspaces   tpssg 24026
            16.11.2  Propositional and predicate calculus   neleq12d 24027
            16.11.3  Linear temporal logic   wbox 24064
            16.11.4  Operations   ssoprab2g 24126
            16.11.5  General Set Theory   uninqs 24133
            16.11.6  The "maps to" notation   cmpfun 24237
            16.11.7  Cartesian Products   cpro 24245
            16.11.8  Operations on subsets and functions   ccst 24267
            16.11.9  Arithmetic   3timesi 24273
            16.11.10  Lattice (algebraic definition)   clatalg 24276
            16.11.11  Currying and Partial Mappings   ccur1 24289
            16.11.12  Order theory (Extensible Structure Builder)   corhom 24302
            16.11.13  Order theory   cpresetrel 24310
            16.11.14  Finite composites ( i. e. finite sums, products ... )   cprd 24393
            16.11.15  Operation properties   ccm1 24426
            16.11.16  Groups and related structures   ridlideq 24430
            16.11.17  Free structures   csubsmg 24478
            16.11.18  Translations   trdom2 24486
            16.11.19  Fields and Rings   com2i 24511
            16.11.20  Ideals   cidln 24538
            16.11.21  Generic modules and vector spaces (New Structure builder)   cact 24542
            16.11.22  Generic modules and vector spaces   cvec 24544
            16.11.23  Real vector spaces   cvr 24584
            16.11.24  Matrices   cmmat 24588
            16.11.25  Affine spaces   craffsp 24594
            16.11.26  Intervals of reals and extended reals   bsi 24596
            16.11.27  Topology   topnem 24607
            16.11.28  Continuous functions   cnrsfin 24620
            16.11.29  Homeomorphisms   dmhmph 24628
            16.11.30  Initial and final topologies   intopcoaconlem3b 24633
            16.11.31  Filters   efilcp 24647
            16.11.32  Limits   plimfil 24653
            16.11.33  Uniform spaces   cunifsp 24680
            16.11.34  Separated spaces: T0, T1, T2 (Hausdorff) ...   hst1 24682
            16.11.35  Compactness   indcomp 24684
            16.11.36  Connectedness   singempcon 24688
            16.11.37  Topological fields   ctopfld 24692
            16.11.38  Standard topology on RR   intrn 24694
            16.11.39  Standard topology of intervals of RR   stoi 24696
            16.11.40  Cantor's set   cntrset 24697
            16.11.41  Pre-calculus and Cartesian geometry   dmse1 24698
            16.11.42  Extended Real numbers   nolimf 24714
            16.11.43  ( RR ^ N ) and ( CC ^ N )   cplcv 24739
            16.11.44  Calculus   cintvl 24791
            16.11.45  Directed multi graphs   cmgra 24803
            16.11.46  Category and deductive system underlying "structure"   calg 24806
            16.11.47  Deductive systems   cded 24829
            16.11.48  Categories   ccatOLD 24847
            16.11.49  Homsets   chomOLD 24880
            16.11.50  Monomorphisms, Epimorphisms, Isomorphisms   cepiOLD 24898
            16.11.51  Functors   cfuncOLD 24926
            16.11.52  Subcategories   csubcat 24938
            16.11.53  Terminal and initial objects   ciobj 24955
            16.11.54  Sources and sinks   csrce 24960
            16.11.55  Limits and co-limits   clmct 24969
            16.11.56  Product and sum of two objects   cprodo 24972
            16.11.57  Tarski's classes   ctar 24976
            16.11.58  Category Set   ccmrcase 25005
            16.11.59  Grammars, Logics, Machines and Automata   ckln 25075
            16.11.60  Words   cwrd 25076
            16.11.61  Planar geometry   cpoints 25151
      16.12  Mathbox for Jeff Hankins
            16.12.1  Miscellany   a1i13 25295
            16.12.2  Basic topological facts   topbnd 25337
            16.12.3  Topology of the real numbers   reconnOLD 25350
            16.12.4  Refinements   cfne 25354
            16.12.5  Neighborhood bases determine topologies   neibastop1 25403
            16.12.6  Lattice structure of topologies   topmtcl 25407
            16.12.7  Filter bases   fgmin 25414
            16.12.8  Directed sets, nets   tailfval 25416
      16.13  Mathbox for Jeff Madsen
            16.13.1  Logic and set theory   anim12da 25427
            16.13.2  Real and complex numbers; integers   fimaxreOLD 25525
            16.13.3  Sequences and sums   sdclem2 25547
            16.13.4  Topology   unopnOLD 25559
            16.13.5  Metric spaces   metf1o 25564
            16.13.6  Continuous maps and homeomorphisms   constcncf 25573
            16.13.7  Product topologies   txtopiOLD 25581
            16.13.8  Boundedness   ctotbnd 25585
            16.13.9  Isometries   cismty 25617
            16.13.10  Heine-Borel Theorem   heibor1lem 25628
            16.13.11  Banach Fixed Point Theorem   bfplem1 25641
            16.13.12  Euclidean space   crrn 25644
            16.13.13  Intervals (continued)   ismrer1 25657
            16.13.14  Groups and related structures   exidcl 25661
            16.13.15  Rings   rngonegcl 25671
            16.13.16  Ring homomorphisms   crnghom 25686
            16.13.17  Commutative rings   ccring 25715
            16.13.18  Ideals   cidl 25727
            16.13.19  Prime rings and integral domains   cprrng 25766
            16.13.20  Ideal generators   cigen 25779
      16.14  Mathbox for Rodolfo Medina
            16.14.1  Partitions   prtlem60 25798
      16.15  Mathbox for Stefan O'Rear
            16.15.1  Additional elementary logic and set theory   nelss 25846
            16.15.2  Additional theory of functions   fninfp 25849
            16.15.3  Extensions beyond function theory   gsumvsmul 25859
            16.15.4  Additional topology   elrfi 25864
            16.15.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 25868
            16.15.6  Algebraic closure systems   cnacs 25872
            16.15.7  Miscellanea 1. Map utilities   constmap 25883
            16.15.8  Miscellanea for polynomials   ofmpteq 25892
            16.15.9  Multivariate polynomials over the integers   cmzpcl 25894
            16.15.10  Miscellanea for Diophantine sets 1   coeq0 25926
            16.15.11  Diophantine sets 1: definitions   cdioph 25929
            16.15.12  Diophantine sets 2 miscellanea   ellz1 25941
            16.15.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 25947
            16.15.14  Diophantine sets 3: construction   diophrex 25950
            16.15.15  Diophantine sets 4 miscellanea   2sbcrex 25959
            16.15.16  Diophantine sets 4: Quantification   rexrabdioph 25970
            16.15.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 25977
            16.15.18  Diophantine sets 6 miscellanea   fz1ssnn 25987
            16.15.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 25989
            16.15.20  Pigeonhole Principle and cardinality helpers   fphpd 25994
            16.15.21  A non-closed set of reals is infinite   rencldnfilem 25998
            16.15.22  Miscellanea for Lagrange's theorem   icodiamlt 26000
            16.15.23  Lagrange's rational approximation theorem   irrapxlem1 26002
            16.15.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26009
            16.15.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26016
            16.15.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26058
            16.15.27  Logarithm laws generalized to an arbitrary base   reglogcl 26070
            16.15.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26078
            16.15.29  X and Y sequences 1: Definition and recurrence laws   crmx 26080
            16.15.30  Ordering and induction lemmas for the integers   monotuz 26121
            16.15.31  X and Y sequences 2: Order properties   rmxypos 26129
            16.15.32  Congruential equations   congtr 26147
            16.15.33  Alternating congruential equations   acongid 26157
            16.15.34  Additional theorems on integer divisibility   bezoutr 26167
            16.15.35  X and Y sequences 3: Divisibility properties   jm2.18 26176
            16.15.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26193
            16.15.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26203
            16.15.38  Uncategorized stuff not associated with a major project   setindtr 26212
            16.15.39  More equivalents of the Axiom of Choice   axac10 26221
            16.15.40  Finitely generated left modules   clfig 26260
            16.15.41  Noetherian left modules I   clnm 26268
            16.15.42  Addenda for structure powers   pwssplit0 26282
            16.15.43  Direct sum of left modules   cdsmm 26292
            16.15.44  Free modules   cfrlm 26307
            16.15.45  Every set admits a group structure iff choice   unxpwdom3 26351
            16.15.46  Independent sets and families   clindf 26369
            16.15.47  Characterization of free modules   lmimlbs 26401
            16.15.48  Noetherian rings and left modules II   clnr 26408
            16.15.49  Hilbert's Basis Theorem   cldgis 26420
            16.15.50  Additional material on polynomials [DEPRECATED]   cmnc 26430
            16.15.51  Degree and minimal polynomial of algebraic numbers   cdgraa 26440
            16.15.52  Algebraic integers I   citgo 26457
            16.15.53  Finite cardinality [SO]   en1uniel 26475
            16.15.54  Words in monoids and ordered group sum   issubmd 26478
            16.15.55  Transpositions in the symmetric group   cpmtr 26479
            16.15.56  The sign of a permutation   cpsgn 26509
            16.15.57  The matrix algebra   cmmul 26534
            16.15.58  The determinant   cmdat 26578
            16.15.59  Endomorphism algebra   cmend 26584
            16.15.60  Subfields   csdrg 26598
            16.15.61  Cyclic groups and order   idomrootle 26606
            16.15.62  Cyclotomic polynomials   ccytp 26616
            16.15.63  Miscellaneous topology   fgraphopab 26624
      16.16  Mathbox for Steve Rodriguez
            16.16.1  Miscellanea   iso0 26631
            16.16.2  Function operations   caofcan 26635
            16.16.3  Calculus   lhe4.4ex1a 26641
      16.17  Mathbox for Andrew Salmon
            16.17.1  Principia Mathematica * 10   pm10.12 26648
            16.17.2  Principia Mathematica * 11   2alanimi 26662
            16.17.3  Predicate Calculus   sbeqal1 26692
            16.17.4  Principia Mathematica * 13 and * 14   pm13.13a 26703
            16.17.5  Set Theory   elnev 26734
            16.17.6  Arithmetic   addcomgi 26757
            16.17.7  Geometry   cplusr 26758
      16.18  Mathbox for Jarvin Udandy
      16.19  Mathbox for David A. Wheeler
            16.19.1  Natural deduction   19.8ad 26805
            16.19.2  Greater than, greater than or equal to.   cge-real 26817
            16.19.3  Hyperbolic trig functions   csinh 26827
            16.19.4  Reciprocal trig functions (sec, csc, cot)   csec 26838
            16.19.5  Identities for "if"   ifnmfalse 26860
            16.19.6  Not-member-of   AnelBC 26861
            16.19.7  Decimal point   cdp2 26862
            16.19.8  Signum (sgn or sign) function   csgn 26870
            16.19.9  Ceiling function   ccei 26880
            16.19.10  Logarithm laws generalized to an arbitrary base   clogb 26884
            16.19.11  Miscellaneous   2m1e1 26889
      16.20  Mathbox for Alan Sare
            16.20.1  Conventional Metamath proofs, some derived from VD proofs   iidn3 26893
            16.20.2  What is Virtual Deduction?   wvd1 26968
            16.20.3  Virtual Deduction Theorems   df-vd1 26969
            16.20.4  Theorems proved using virtual deduction   trsspwALT 27220
            16.20.5  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 27250
            16.20.6  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 27317
            16.20.7  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 27321
            16.20.8  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 27328
            16.20.9  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 27331
      16.21  Mathbox for Jonathan Ben-Naim
            16.21.1  First order logic and set theory   bnj170 27350
            16.21.2  Well founded induction and recursion   bnj110 27517
            16.21.3  The existence of a minimal element in certain classes   bnj69 27667
            16.21.4  Well-founded induction   bnj1204 27669
            16.21.5  Well-founded recursion, part 1 of 3   bnj60 27719
            16.21.6  Well-founded recursion, part 2 of 3   bnj1500 27725
            16.21.7  Well-founded recursion, part 3 of 3   bnj1522 27729
      16.22  Mathbox for Norm Megill
            16.22.1  Obsolete experiments to study ax-12o   ax12-2 27730
            16.22.2  Miscellanea   cnaddcom 27788
            16.22.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 27791
            16.22.4  Functionals and kernels of a left vector space (or module)   clfn 27874
            16.22.5  Opposite rings and dual vector spaces   cld 27940
            16.22.6  Ortholattices and orthomodular lattices   cops 27989
            16.22.7  Atomic lattices with covering property   ccvr 28079
            16.22.8  Hilbert lattices   chlt 28167
            16.22.9  Projective geometries based on Hilbert lattices   clln 28307
            16.22.10  Construction of a vector space from a Hilbert lattice   cdlema1N 28607
            16.22.11  Construction of involution and inner product from a Hilbert lattice   clpoN 30297

    < 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 30701-30781
  Copyright terms: Public domain < Wrap  Next >