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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
2.2  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
16.14  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

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
1.4  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
2.1.1  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 2509
2.1.6  The universal class   cvv 2727
2.1.7  Conditional equality (experimental)   wcdeq 2904
2.1.9  Proper substitution of classes for sets   wsbc 2921
2.1.10  Proper substitution of classes for sets into classes   csb 3009
2.1.11  Define basic set operations and relations   cdif 3075
2.1.12  Subclasses and subsets   df-ss 3089
2.1.13  The difference, union, and intersection of two classes   difeq1 3204
2.1.14  The empty set   c0 3362
2.1.15  "Weak deduction theorem" for set theory   cif 3470
2.1.16  Power classes   cpw 3530
2.1.17  Unordered and ordered pairs   csn 3544
2.1.18  The union of a class   cuni 3727
2.1.19  The intersection of a class   cint 3760
2.1.20  Indexed union and intersection   ciun 3803
2.1.21  Disjointness   wdisj 3891
2.1.22  Binary relations   wbr 3920
2.1.23  Ordered-pair class abstractions (class builders)   copab 3973
2.1.24  Transitive classes   wtr 4010
2.2  ZF Set Theory - add the Axiom of Replacement
2.2.1  Introduce the Axiom of Replacement   ax-rep 4028
2.2.2  Derive the Axiom of Separation   axsep 4037
2.2.3  Derive the Null Set Axiom   zfnuleu 4043
2.2.4  Theorems requiring subset and intersection existence   nalset 4048
2.2.5  Theorems requiring empty set existence   class2set 4072
2.3  ZF Set Theory - add the Axiom of Power Sets
2.3.1  Introduce the Axiom of Power Sets   ax-pow 4082
2.3.2  Derive the Axiom of Pairing   zfpair 4106
2.3.3  Ordered pair theorem   opnz 4135
2.3.4  Ordered-pair class abstractions (cont.)   opabid 4164
2.3.5  Power class of union and intersection   pwin 4190
2.3.6  Epsilon and identity relations   cep 4196
2.3.7  Partial and complete ordering   wpo 4205
2.3.8  Founded and well-ordering relations   wfr 4242
2.3.9  Ordinals   word 4284
2.4  ZF Set Theory - add the Axiom of Union
2.4.1  Introduce the Axiom of Union   ax-un 4403
2.4.2  Ordinals (continued)   ordon 4465
2.4.3  Transfinite induction   tfi 4535
2.4.4  The natural numbers (i.e. finite ordinals)   com 4547
2.4.5  Peano's postulates   peano1 4566
2.4.6  Finite induction (for finite ordinals)   find 4572
2.4.7  Functions and relations   cxp 4578
2.4.8  Operations   co 5710
2.4.9  "Maps to" notation   elmpt2cl 5913
2.4.10  Function operation   cof 5928
2.4.11  First and second members of an ordered pair   c1st 5972
2.4.12  Function transposition   ctpos 6085
2.4.13  Curry and uncurry   ccur 6124
2.4.14  Proper subset relation   crpss 6128
2.4.15  Definite description binder (inverted iota)   cio 6141
2.4.16  Cantor's Theorem   canth 6178
2.4.17  Undefined values and restricted iota (description binder)   cund 6180
2.4.18  Functions on ordinals; strictly monotone ordinal functions   iunon 6241
2.4.19  "Strong" transfinite recursion   crecs 6273
2.4.20  Recursive definition generator   crdg 6308
2.4.21  Finite recursion   frfnom 6333
2.4.22  Abian's "most fundamental" fixed point theorem   abianfplem 6356
2.4.23  Ordinal arithmetic   c1o 6358
2.4.24  Natural number arithmetic   nna0 6488
2.4.25  Equivalence relations and classes   wer 6543
2.4.26  The mapping operation   cmap 6658
2.4.27  Infinite Cartesian products   cixp 6703
2.4.28  Equinumerosity   cen 6746
2.4.29  Schroeder-Bernstein Theorem   sbthlem1 6856
2.4.30  Equinumerosity (cont.)   xpf1o 6908
2.4.31  Pigeonhole Principle   phplem1 6925
2.4.32  Finite sets   onomeneq 6935
2.4.33  Finite intersections   cfi 7048
2.4.34  Hall's marriage theorem   marypha1lem 7070
2.4.35  Supremum   csup 7077
2.4.36  Ordinal isomorphism, Hartog's theorem   coi 7108
2.4.37  Hartogs function, order types, weak dominance   char 7154
2.5  ZF Set Theory - add the Axiom of Regularity
2.5.1  Introduce the Axiom of Regularity   ax-reg 7190
2.5.2  Axiom of Infinity equivalents   inf0 7206
2.6  ZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity   ax-inf 7223
2.6.2  Existence of omega (the set of natural numbers)   omex 7228
2.6.3  Cantor normal form   ccnf 7246
2.6.4  Transitive closure   trcl 7294
2.6.5  Rank   cr1 7318
2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7439
2.6.7  Cardinal numbers   ccrd 7452
2.6.8  Axiom of Choice equivalents   wac 7626
2.6.9  Cardinal number arithmetic   ccda 7677
2.6.10  The Ackermann bijection   ackbij2lem1 7729
2.6.11  Cofinality (without Axiom of Choice)   cflem 7756
2.6.12  Eight inequivalent definitions of finite set   sornom 7787
2.6.13  Hereditarily size-limited sets without Choice   itunifval 7926
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 7969
3.2.2  AC equivalents: well ordering, Zorn's lemma   numthcor 8005
3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8052
3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8076
3.2.5  Cofinality using Axiom of Choice   alephreg 8084
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 8184
4.1.2  Weak universes   cwun 8202
4.1.3  Tarski's classes   ctsk 8250
4.1.4  Grothendieck's universes   cgru 8292
4.2  ZFC Set Theory plus the Tarksi-Grothendieck Axiom
4.2.1  Introduce the Tarksi-Grothendieck Axiom   ax-groth 8325
4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8328
4.2.3  Tarski map function   ctskm 8339
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 8346
5.1.2  Final derivation of real and complex number postulates   axaddf 8647
5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8673
5.2  Derive the basic properties from the field axioms
5.2.1  Some deductions from the field axioms for complex numbers   cnex 8698
5.2.2  Infinity and the extended real number system   cpnf 8744
5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 8774
5.2.4  Ordering on reals   lttr 8779
5.2.5  Initial properties of the complex numbers   mul12 8858
5.3  Real and complex numbers - basic operations
5.3.2  Subtraction   cmin 8917
5.3.4  Ordering on reals (cont.)   gt0ne0 9119
5.3.5  Reciprocals   ixi 9277
5.3.6  Division   cdiv 9303
5.3.7  Ordering on reals (cont.)   elimgt0 9472
5.3.8  Completeness Axiom and Suprema   fimaxre 9581
5.3.9  Imaginary and complex number properties   inelr 9616
5.3.10  Function operation analogue theorems   ofsubeq0 9623
5.4  Integer sets
5.4.1  Natural numbers (as a subset of complex numbers)   cn 9626
5.4.2  Principle of mathematical induction   nnind 9644
5.4.3  Decimal representation of numbers   c2 9675
5.4.4  Some properties of specific numbers   0p1e1 9719
5.4.5  The Archimedean property   nnunb 9840
5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 9844
5.4.7  Integers (as a subset of complex numbers)   cz 9903
5.4.8  Decimal arithmetic   cdc 10003
5.4.9  Upper partititions of integers   cuz 10109
5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10190
5.4.11  Rational numbers (as a subset of complex numbers)   cq 10195
5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10221
5.5  Order sets
5.5.1  Positive reals (as a subset of complex numbers)   crp 10233
5.5.2  Infinity and the extended real number system (cont.)   cxne 10328
5.5.3  Supremum on the extended reals   xrsupexmnf 10501
5.5.4  Real number intervals   cioo 10534
5.5.5  Finite intervals of integers   cfz 10660
5.5.6  Half-open integer ranges   cfzo 10748
5.6  Elementary integer functions
5.6.1  The floor (greatest integer) function   cfl 10802
5.6.2  The modulo (remainder) operation   cmo 10851
5.6.3  The infinite sequence builder "seq"   om2uz0i 10888
5.6.4  Integer powers   cexp 10982
5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11160
5.6.6  Factorial function   cfa 11166
5.6.7  The binomial coefficient operation   cbc 11193
5.6.8  The ` # ` (finite set size) function   chash 11215
5.6.9  Words over a set   cword 11280
5.6.10  Longer string literals   cs2 11368
5.7  Elementary real and complex functions
5.7.1  The "shift" operation   cshi 11438
5.7.2  Real and imaginary parts; conjugate   ccj 11458
5.7.3  Square root; absolute value   csqr 11595
5.8  Elementary limits and convergence
5.8.1  Superior limit (lim sup)   clsp 11821
5.8.2  Limits   cli 11835
5.8.3  Finite and infinite sums   csu 12035
5.8.4  The binomial theorem   binomlem 12164
5.8.5  Infinite sums (cont.)   isumshft 12172
5.8.6  Miscellaneous converging and diverging sequences   divrcnv 12185
5.8.7  Arithmetic series   arisum 12192
5.8.8  Geometric series   expcnv 12196
5.8.9  Ratio test for infinite series convergence   cvgrat 12213
5.8.10  Mertens' theorem   mertenslem1 12214
5.9  Elementary trigonometry
5.9.1  The exponential, sine, and cosine functions   ce 12217
5.9.2  _e is irrational   eirrlem 12356
5.10  Cardinality of real and complex number subsets
5.10.1  Countability of integers and rationals   xpnnen 12361
5.10.2  The reals are uncountable   rpnnen2lem1 12367
PART 6  ELEMENTARY NUMBER THEORY
6.1  Elementary properties of divisibility
6.1.1  Irrationality of square root of 2   sqr2irrlem 12400
6.1.2  Some Number sets are chains of proper subsets   nthruc 12403
6.1.3  The divides relation   cdivides 12405
6.1.4  The division algorithm   divalglem0 12466
6.1.5  Bit sequences   cbits 12484
6.1.6  The greatest common divisor operator   cgcd 12559
6.1.7  Bézout's identity   bezoutlem1 12591
6.1.8  Algorithms   nn0seqcvgd 12614
6.1.9  Euclid's Algorithm   eucalgval2 12625
6.2  Elementary prime number theory
6.2.1  Elementary properties   cprime 12632
6.2.2  Properties of the canonical representation of a rational   cnumer 12678
6.2.3  Euler's theorem   codz 12705
6.2.4  Pythagorean Triples   coprimeprodsq 12736
6.2.5  The prime count function   cpc 12763
6.2.6  Pocklington's theorem   prmpwdvds 12825
6.2.7  Infinite primes theorem   unbenlem 12829
6.2.8  Sum of prime reciprocals   prmreclem1 12837
6.2.9  Fundamental theorem of arithmetic   1arithlem1 12844
6.2.10  Lagrange's four-square theorem   cgz 12850
6.2.11  Van der Waerden's theorem   cvdwa 12886
6.2.12  Ramsey's theorem   cram 12920
6.2.13  Decimal arithmetic (cont.)   dec2dvds 12952
6.2.14  Specific prime numbers   4nprm 12980
6.2.15  Very large primes   1259lem1 13003
PART 7  EXTENSIBLE STRUCTURES
7.1  Extensible structures
7.1.1  Basic definitions   cstr 13018
7.1.2  Slot definitions   cplusg 13082
7.1.3  Definition of the structure product   crest 13199
7.1.4  Definition of the structure quotient   cordt 13272
7.2  Moore spaces
7.2.1  Moore closures   mrcflem 13380
7.2.2  Algebraic closure systems   isacs 13398
PART 8  BASIC CATEGORY THEORY
8.1  Categories
8.1.1  Categories   ccat 13410
8.1.2  Opposite category   coppc 13458
8.1.3  Monomorphisms and epimorphisms   cmon 13475
8.1.4  Sections, inverses, isomorphisms   csect 13491
8.1.5  Subcategories   cssc 13528
8.1.6  Functors   cfunc 13572
8.1.7  Full & faithful functors   cful 13620
8.1.8  Natural transformations and the functor category   cnat 13659
8.2  Arrows (disjointified hom-sets)
8.2.1  Identity and composition for arrows   cida 13729
8.3  Examples of categories
8.3.1  The category of sets   csetc 13751
8.3.2  The category of categories   ccatc 13770
8.4  Categorical constructions
8.4.1  Product of categories   cxpc 13786
8.4.2  Functor evaluation   cevlf 13827
8.4.3  Hom functor   chof 13866
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 13918
9.2.2  Lattices   clat 13995
9.2.3  The dual of an ordered set   codu 14076
9.2.4  Subset order structures   cipo 14098
9.2.5  Distributive lattices   latmass 14126
9.2.6  Posets and lattices as relations   cps 14136
9.2.7  Directed sets, nets   cdir 14185
PART 10  BASIC ALGEBRAIC STRUCTURES
10.1  Monoids
10.1.1  Definition and basic properties   cmnd 14196
10.1.2  Monoid homomorphisms and submonoids   cmhm 14248
10.1.3  Ordered group sum operation   gsumvallem1 14283
10.1.4  Free monoids   cfrmd 14304
10.2  Groups
10.2.1  Definition and basic properties   df-grp 14324
10.2.2  Subgroups and Quotient groups   csubg 14450
10.2.3  Elementary theory of group homomorphisms   cghm 14515
10.2.4  Isomorphisms of groups   cgim 14556
10.2.5  Group actions   cga 14578
10.2.6  Symmetry groups and Cayley's Theorem   csymg 14604
10.2.7  Centralizers and centers   ccntz 14626
10.2.8  The opposite group   coppg 14653
10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 14675
10.2.10  Direct products   clsm 14780
10.2.11  Free groups   cefg 14850
10.3  Abelian groups
10.3.1  Definition and basic properties   ccmn 14924
10.3.2  Cyclic groups   ccyg 14999
10.3.3  Group sum operation   gsumval3a 15024
10.3.4  Internal direct products   cdprd 15066
10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15135
10.4  Rings
10.4.1  Multiplicative Group   cmgp 15160
10.4.2  Definition and basic properties   crg 15172
10.4.3  Opposite ring   coppr 15239
10.4.4  Divisibility   cdsr 15255
10.4.5  Ring homomorphisms   crh 15329
10.5  Division rings and Fields
10.5.1  Definition and basic properties   cdr 15347
10.5.2  Subrings of a ring   csubrg 15376
10.5.3  Absolute value (abstract algebra)   cabv 15416
10.5.4  Star rings   cstf 15443
10.6  Left Modules
10.6.1  Definition and basic properties   clmod 15462
10.6.2  Subspaces and spans in a left module   clss 15524
10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 15611
10.6.4  Subspace sum; bases for a left module   clbs 15662
10.7  Vector Spaces
10.7.1  Definition and basic properties   clvec 15690
10.8  Ideals
10.8.1  The subring algebra; ideals   csra 15753
10.8.2  Two-sided ideals and quotient rings   c2idl 15815
10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 15825
10.8.4  Nonzero rings   cnzr 15841
10.8.5  Left regular elements. More kinds of ring   crlreg 15852
10.9  Associative algebras
10.9.1  Definition and basic properties   casa 15882
10.10  Abstract Multivariate Polynomials
10.10.1  Definition and basic properties   cmps 15919
10.10.2  Polynomial evaluation   evlslem4 16077
10.10.3  Univariate Polynomials   cps1 16082
10.11  The complex numbers as an extensible structure
10.11.1  Definition and basic properties   cxmt 16201
10.11.2  Algebraic constructions based on the complexes   czrh 16283
10.12  Hilbert spaces
10.12.1  Definition and basic properties   cphl 16360
10.12.2  Orthocomplements and closed subspaces   cocv 16392
10.12.3  Orthogonal projection and orthonormal bases   cpj 16432
PART 11  BASIC TOPOLOGY
11.1  Topology
11.1.1  Topological spaces   ctop 16463
11.1.2  TopBases for topologies   isbasisg 16517
11.1.3  Examples of topologies   distop 16565
11.1.4  Closure and interior   ccld 16585
11.1.5  Neighborhoods   cnei 16666
11.1.6  Limit points and perfect sets   clp 16698
11.1.7  Subspace topologies   restrcl 16720
11.1.8  Order topology   ordtbaslem 16750
11.1.9  Limits and Continuity in topological spaces   ccn 16786
11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 16866
11.1.11  Compactness   ccmp 16945
11.1.12  Connectedness   ccon 16969
11.1.13  First- and second-countability   c1stc 16995
11.1.14  Local topological properties   clly 17022
11.1.15  Compactly generated spaces   ckgen 17060
11.1.16  Product topologies   ctx 17087
11.1.17  Continuous function-builders   cnmptid 17187
11.1.18  Quotient maps and quotient topology   ckq 17216
11.1.19  Homeomorphisms   chmeo 17276
11.2  Filters and filter bases
11.2.1  Filter Bases   cfbas 17350
11.2.2  Filters   cfil 17372
11.2.3  Ultrafilters   cufil 17426
11.2.4  Filter limits   cfm 17460
11.2.5  Topological groups   ctmd 17585
11.2.6  Infinite group sum on topological groups   ctsu 17640
11.2.7  Topological rings, fields, vector spaces   ctrg 17670
11.3  Metric spaces
11.3.1  Basic metric space properties   cxme 17714
11.3.2  Metric space balls   blfval 17779
11.3.3  Open sets of a metric space   mopnval 17816
11.3.4  Continuity in metric spaces   metcnp3 17918
11.3.5  Examples of metric spaces   dscmet 17927
11.3.6  Normed algebraic structures   cnm 17931
11.3.7  Normed space homomorphisms (bounded linear operators)   cnmo 18046
11.3.8  Topology on the Reals   qtopbaslem 18099
11.3.9  Topological definitions using the reals   cii 18211
11.3.10  Path homotopy   chtpy 18297
11.3.11  The fundamental group   cpco 18330
11.3.12  Complex left modules   cclm 18392
11.3.13  Complex pre-Hilbert space   ccph 18434
11.3.14  Convergence and completeness   ccfil 18510
11.3.15  Baire's Category Theorem   bcthlem1 18578
11.3.16  Banach spaces and complex Hilbert spaces   ccms 18586
11.3.17  Minimizing Vector Theorem   minveclem1 18620
11.3.18  Projection theorem   pjthlem1 18633
PART 12  BASIC REAL AND COMPLEX ANALYSIS
12.1  Continuity
12.1.1  Intermediate value theorem   pmltpclem1 18640
12.2  Integrals
12.2.1  Lebesgue measure   covol 18654
12.2.2  Lebesgue integration   cmbf 18801
12.3  Derivatives
12.3.1  Real and Complex Differentiation   climc 19044
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
13.1  Polynomials
13.1.1  Abstract polynomials, continued   evlslem6 19229
13.1.2  Polynomial degrees   cmdg 19271
13.1.3  The division algorithm for univariate polynomials   cmn1 19343
13.1.4  Elementary properties of complex polynomials   cply 19398
13.1.5  The Division algorithm for polynomials   cquot 19502
13.1.6  Algebraic numbers   caa 19526
13.1.7  Liouville's approximation theorem   aalioulem1 19544
13.2  Sequences and series
13.2.1  Taylor polynomials and Taylor's theorem   ctayl 19564
13.2.2  Uniform convergence   culm 19587
13.2.3  Power series   pserval 19618
13.3  Basic trigonometry
13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 19651
13.3.2  Properties of pi = 3.14159...   pilem1 19659
13.3.3  Mapping of the exponential function   efgh 19735
13.3.4  The natural logarithm on complex numbers   clog 19744
13.3.5  Solutions of quardatic, cubic, and quartic equations   quad2 19967
13.3.6  Inverse trigonometric functions   casin 19990
13.3.7  The Birthday Problem   log2ublem1 20074
13.3.8  Areas in R^2   carea 20082
13.3.9  More miscellaneous converging sequences   rlimcnp 20092
13.3.10  Inequality of arithmetic and geometric means   cvxcl 20111
13.3.11  Euler-Mascheroni constant   cem 20118
13.4  Basic number theory
13.4.1  Wilson's theorem   wilthlem1 20138
13.4.2  The Fundamental Theorem of Algebra   ftalem1 20142
13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20150
13.4.4  Number-theoretical functions   ccht 20160
13.4.5  Perfect Number Theorem   mersenne 20298
13.4.6  Characters of Z/nZ   cdchr 20303
13.4.7  Bertrand's postulate   bcctr 20346
13.4.8  Legendre symbol   clgs 20365
13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 20434
13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 20450
13.4.12  The Prime Number Theorem   mudivsum 20511
13.4.13  Ostrowski's theorem   abvcxp 20596
PART 14  MISCELLANEA
14.1  Definitional Examples
14.2  Natural deduction examples
14.3  Humor
14.3.1  April Fool's theorem   avril1 20666
14.4  (Future - to be reviewed and classified)
14.4.1  Planar incidence geometry   cplig 20672
14.4.2  Algebra preliminaries   crpm 20677
14.4.3  Transitive closure   ctcl 20679
PART 15  DEPRECATED SECTIONS
15.1  Additional material on Group theory
15.1.1  Definitions and basic properties for groups   cgr 20683
15.1.2  Definition and basic properties of Abelian groups   cablo 20778
15.1.3  Subgroups   csubgo 20798
15.1.4  Operation properties   cass 20809
15.1.5  Group-like structures   cmagm 20815
15.1.6  Examples of Abelian groups   ablosn 20844
15.1.7  Group homomorphism and isomorphism   cghom 20854
15.2  Additional material on Rings and Fields
15.2.1  Definition and basic properties   crngo 20872
15.2.2  Examples of rings   cnrngo 20900
15.2.3  Division Rings   cdrng 20902
15.2.4  Star Fields   csfld 20905
15.2.5  Fields and Rings   ccm2 20907
15.3  Complex vector spaces
15.3.1  Definition and basic properties   cvc 20931
15.3.2  Examples of complex vector spaces   cncvc 20969
15.4  Normed complex vector spaces
15.4.1  Definition and basic properties   cnv 20970
15.4.2  Examples of normed complex vector spaces   cnnv 21075
15.4.3  Induced metric of a normed complex vector space   imsval 21084
15.4.4  Inner product   cdip 21103
15.4.5  Subspaces   css 21127
15.5  Operators on complex vector spaces
15.5.1  Definitions and basic properties   clno 21148
15.6  Inner product (pre-Hilbert) spaces
15.6.1  Definition and basic properties   ccphlo 21220
15.6.2  Examples of pre-Hilbert spaces   cncph 21227
15.6.3  Properties of pre-Hilbert spaces   isph 21230
15.7  Complex Banach spaces
15.7.1  Definition and basic properties   ccbn 21271
15.7.2  Examples of complex Banach spaces   cnbn 21278
15.7.3  Uniform Boundedness Theorem   ubthlem1 21279
15.7.4  Minimizing Vector Theorem   minvecolem1 21283
15.8  Complex Hilbert spaces
15.8.1  Definition and basic properties   chlo 21294
15.8.2  Standard axioms for a complex Hilbert space   hlex 21307
15.8.3  Examples of complex Hilbert spaces   cnchl 21325
15.8.4  Subspaces   ssphl 21326
15.8.5  Hellinger-Toeplitz Theorem   htthlem 21327
15.9  Hilbert Space Explorer
15.9.1  Basic Hilbert space definitions   chil 21329
15.9.2  Preliminary ZFC lemmas   df-hnorm 21378
15.9.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 21391
15.9.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 21409
15.9.5  Vector operations   hvmulex 21421
15.9.6  Inner product postulates for a Hilbert space   ax-hfi 21488
15.9.7  Inner product   his5 21495
15.9.8  Norms   dfhnorm2 21531
15.9.9  Relate Hilbert space to normed complex vector spaces   hilablo 21569
15.9.11  Cauchy sequences and limits   hcau 21593
15.9.12  Derivation of the completeness axiom from ZF set theory   hilmet 21603
15.9.13  Completeness postulate for a Hilbert space   ax-hcompl 21611
15.9.14  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 21612
15.9.15  Subspaces   df-sh 21616
15.9.16  Closed subspaces   df-ch 21631
15.9.17  Orthocomplements   df-oc 21661
15.9.18  Subspace sum, span, lattice join, lattice supremum   df-shs 21717
15.9.19  Projection theorem   pjhthlem1 21800
15.9.20  Projectors   df-pjh 21804
15.9.21  Orthomodular law   omlsilem 21811
15.9.22  Projectors (cont.)   pjhtheu2 21825
15.9.23  Hilbert lattice operations   sh0le 21849
15.9.24  Span (cont.) and one-dimensional subspaces   spansn0 21950
15.9.25  Operator sum, difference, and scalar multiplication   df-hosum 21992
15.9.26  Commutes relation for Hilbert lattice elements   df-cm 22010
15.9.27  Foulis-Holland theorem   fh1 22045
15.9.28  Quantum Logic Explorer axioms   qlax1i 22054
15.9.29  Orthogonal subspaces   chscllem1 22064
15.9.30  Orthoarguesian laws 5OA and 3OA   5oalem1 22081
15.9.31  Projectors (cont.)   pjorthi 22096
15.9.32  Mayet's equation E_3   mayete3i 22155
15.9.33  Zero and identity operators   df-h0op 22158
15.9.34  Operations on Hilbert space operators   hoaddcl 22168
15.9.35  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 22249
15.9.36  Linear and continuous functionals and norms   df-nmfn 22255
15.9.38  Dirac bra-ket notation   df-bra 22260
15.9.39  Positive operators   df-leop 22262
15.9.40  Eigenvectors, eigenvalues, spectrum   df-eigvec 22263
15.9.41  Theorems about operators and functionals   nmopval 22266
15.9.42  Riesz lemma   riesz3i 22472
15.9.44  Quantum computation error bound theorem   unierri 22514
15.9.45  Dirac bra-ket notation (cont.)   branmfn 22515
15.9.46  Positive operators (cont.)   leopg 22532
15.9.47  Projectors as operators   pjhmopi 22556
15.9.48  States on a Hilbert lattice   df-st 22621
15.9.49  Godowski's equation   golem1 22681
15.9.50  Covers relation; modular pairs   df-cv 22689
15.9.51  Atoms   df-at 22748
15.9.52  Superposition principle   superpos 22764
15.9.53  Atoms, exchange and covering properties, atomicity   chcv1 22765
15.9.54  Irreducibility   chirredlem1 22800
15.9.55  Atoms (cont.)   atcvat3i 22806
15.9.56  Modular symmetry   mdsymlem1 22813
PART 16  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
16.1  Mathboxes for user contributions
16.1.1  Mathbox guidelines   mathbox 22852
16.2  Mathbox for Stefan Allan
16.3  Mathbox for Mario Carneiro
16.3.1  Miscellaneous stuff   quartfull 22857
16.3.2  Zeta function   czeta 22858
16.3.3  Gamma function   clgam 22861
16.3.4  Derangements and the Subfactorial   deranglem 22868
16.3.5  The Erdős-Szekeres theorem   erdszelem1 22893
16.3.6  The Kuratowski closure-complement theorem   kur14lem1 22908
16.3.7  Retracts and sections   cretr 22919
16.3.8  Path-connected and simply connected spaces   cpcon 22921
16.3.9  Covering maps   ccvm 22957
16.3.10  Undirected multigraphs   cumg 23031
16.3.11  Normal numbers   snmlff 23083
16.3.12  Godel-sets of formulas   cgoe 23087
16.3.13  Models of ZF   cgze 23115
16.3.14  Splitting fields   citr 23129
16.3.15  p-adic number fields   czr 23145
16.4  Mathbox for Paul Chapman
16.4.1  Group homomorphism and isomorphism   ghomgrpilem1 23163
16.4.2  Real and complex numbers (cont.)   climuzcnv 23175
16.4.3  Miscellaneous theorems   elfzm12 23179
16.5  Mathbox for Drahflow
16.6  Mathbox for Scott Fenton
16.6.1  ZFC Axioms in primitive form   axextprim 23218
16.6.2  Untangled classes   untelirr 23225
16.6.3  Extra propositional calculus theorems   3orel1 23232
16.6.4  Misc. Useful Theorems   nepss 23243
16.6.5  Properties of reals and complexes   sqdivzi 23249
16.6.6  Greatest common divisor and divisibility   pdivsq 23272
16.6.7  Properties of relationships   brtp 23276
16.6.8  Properties of functions and mappings   funpsstri 23289
16.6.9  Epsilon induction   setinds 23302
16.6.10  Ordinal numbers   elpotr 23305
16.6.11  Defined equality axioms   axextdfeq 23322
16.6.12  Hypothesis builders   hbntg 23330
16.6.13  The Predecessor Class   cpred 23335
16.6.14  (Trans)finite Recursion Theorems   tfisg 23372
16.6.15  Well-founded induction   tz6.26 23373
16.6.16  Transitive closure under a relationship   ctrpred 23388
16.6.17  Founded Induction   frmin 23410
16.6.18  Ordering Ordinal Sequences   orderseqlem 23420
16.6.19  Well-founded recursion   wfr3g 23423
16.6.20  Transfinite recursion via Well-founded recursion   tfrALTlem 23444
16.6.21  Founded Recursion   frr3g 23448
16.6.22  Surreal Numbers   csur 23462
16.6.23  Surreal Numbers: Ordering   axsltsolem1 23489
16.6.24  Surreal Numbers: Birthday Function   axbday 23496
16.6.25  Surreal Numbers: Density   axdenselem1 23503
16.6.26  Surreal Numbers: Full-Eta Property   axfelem1 23514
16.6.27  Symmetric difference   csymdif 23536
16.6.28  Quantifier-free definitions   ctxp 23548
16.6.29  Alternate ordered pairs   caltop 23664
16.6.30  Tarskian geometry   cee 23690
16.6.31  Tarski's axioms for geometry   axdimuniq 23715
16.6.32  Congruence properties   cofs 23779
16.6.33  Betweenness properties   btwntriv2 23809
16.6.34  Segment Transportation   ctransport 23826
16.6.35  Properties relating betweenness and congruence   cifs 23832
16.6.36  Connectivity of betweenness   btwnconn1lem1 23884
16.6.37  Segment less than or equal to   csegle 23903
16.6.38  Outside of relationship   coutsideof 23916
16.6.39  Lines and Rays   cline2 23931
16.6.40  Bernoulli polynomials and sums of k-th powers   cbp 23955
16.6.41  Rank theorems   rankung 23970
16.6.42  Hereditarily Finite Sets   chf 23976
16.7  Mathbox for Anthony Hart
16.7.1  Propositional Calculus   tb-ax1 23991
16.7.2  Predicate Calculus   quantriv 24013
16.7.3  Misc. Single Axiom Systems   meran1 24024
16.7.4  Connective Symmetry   negsym1 24030
16.8  Mathbox for Chen-Pang He
16.8.1  Ordinal topology   ontopbas 24041
16.9  Mathbox for Jeff Hoffman
16.9.1  Inferences for finite induction on generic function values   fveleq 24064
16.9.2  gdc.mm   nnssi2 24068
16.10  Mathbox for Wolf Lammen
16.11  Mathbox for Frédéric Liné
16.11.1  Theorems from other workspaces   tpssg 24097
16.11.2  Propositional and predicate calculus   neleq12d 24098
16.11.3  Linear temporal logic   wbox 24135
16.11.4  Operations   ssoprab2g 24197
16.11.5  General Set Theory   uninqs 24204
16.11.6  The "maps to" notation   cmpfun 24308
16.11.7  Cartesian Products   cpro 24316
16.11.8  Operations on subsets and functions   ccst 24338
16.11.9  Arithmetic   3timesi 24344
16.11.10  Lattice (algebraic definition)   clatalg 24347
16.11.11  Currying and Partial Mappings   ccur1 24360
16.11.12  Order theory (Extensible Structure Builder)   corhom 24373
16.11.13  Order theory   cpresetrel 24381
16.11.14  Finite composites ( i. e. finite sums, products ... )   cprd 24464
16.11.15  Operation properties   ccm1 24497
16.11.16  Groups and related structures   ridlideq 24501
16.11.17  Free structures   csubsmg 24549
16.11.18  Translations   trdom2 24557
16.11.19  Fields and Rings   com2i 24582
16.11.20  Ideals   cidln 24609
16.11.21  Generic modules and vector spaces (New Structure builder)   cact 24613
16.11.22  Generic modules and vector spaces   cvec 24615
16.11.23  Real vector spaces   cvr 24655
16.11.24  Matrices   cmmat 24659
16.11.25  Affine spaces   craffsp 24665
16.11.26  Intervals of reals and extended reals   bsi 24667
16.11.27  Topology   topnem 24678
16.11.28  Continuous functions   cnrsfin 24691
16.11.29  Homeomorphisms   dmhmph 24699
16.11.30  Initial and final topologies   intopcoaconlem3b 24704
16.11.31  Filters   efilcp 24718
16.11.32  Limits   plimfil 24724
16.11.33  Uniform spaces   cunifsp 24751
16.11.34  Separated spaces: T0, T1, T2 (Hausdorff) ...   hst1 24753
16.11.35  Compactness   indcomp 24755
16.11.36  Connectedness   singempcon 24759
16.11.37  Topological fields   ctopfld 24763
16.11.38  Standard topology on RR   intrn 24765
16.11.39  Standard topology of intervals of RR   stoi 24767
16.11.40  Cantor's set   cntrset 24768
16.11.41  Pre-calculus and Cartesian geometry   dmse1 24769
16.11.42  Extended Real numbers   nolimf 24785
16.11.43  ( RR ^ N ) and ( CC ^ N )   cplcv 24810
16.11.44  Calculus   cintvl 24862
16.11.45  Directed multi graphs   cmgra 24874
16.11.46  Category and deductive system underlying "structure"   calg 24877
16.11.47  Deductive systems   cded 24900
16.11.48  Categories   ccatOLD 24918
16.11.49  Homsets   chomOLD 24951
16.11.50  Monomorphisms, Epimorphisms, Isomorphisms   cepiOLD 24969
16.11.51  Functors   cfuncOLD 24997
16.11.52  Subcategories   csubcat 25009
16.11.53  Terminal and initial objects   ciobj 25026
16.11.54  Sources and sinks   csrce 25031
16.11.55  Limits and co-limits   clmct 25040
16.11.56  Product and sum of two objects   cprodo 25043
16.11.57  Tarski's classes   ctar 25047
16.11.58  Category Set   ccmrcase 25076
16.11.59  Grammars, Logics, Machines and Automata   ckln 25146
16.11.60  Words   cwrd 25147
16.11.61  Planar geometry   cpoints 25222
16.12  Mathbox for Jeff Hankins
16.12.1  Miscellany   a1i13 25366
16.12.2  Basic topological facts   topbnd 25408
16.12.3  Topology of the real numbers   reconnOLD 25421
16.12.4  Refinements   cfne 25425
16.12.5  Neighborhood bases determine topologies   neibastop1 25474
16.12.6  Lattice structure of topologies   topmtcl 25478
16.12.7  Filter bases   fgmin 25485
16.12.8  Directed sets, nets   tailfval 25487
16.13.1  Logic and set theory   anim12da 25498
16.13.2  Real and complex numbers; integers   fimaxreOLD 25596
16.13.3  Sequences and sums   sdclem2 25618
16.13.4  Topology   unopnOLD 25630
16.13.5  Metric spaces   metf1o 25635
16.13.6  Continuous maps and homeomorphisms   constcncf 25644
16.13.7  Product topologies   txtopiOLD 25652
16.13.8  Boundedness   ctotbnd 25656
16.13.9  Isometries   cismty 25688
16.13.10  Heine-Borel Theorem   heibor1lem 25699
16.13.11  Banach Fixed Point Theorem   bfplem1 25712
16.13.12  Euclidean space   crrn 25715
16.13.13  Intervals (continued)   ismrer1 25728
16.13.14  Groups and related structures   exidcl 25732
16.13.15  Rings   rngonegcl 25742
16.13.16  Ring homomorphisms   crnghom 25757
16.13.17  Commutative rings   ccring 25786
16.13.18  Ideals   cidl 25798
16.13.19  Prime rings and integral domains   cprrng 25837
16.13.20  Ideal generators   cigen 25850
16.14  Mathbox for Rodolfo Medina
16.14.1  Partitions   prtlem60 25869
16.15  Mathbox for Stefan O'Rear
16.15.1  Additional elementary logic and set theory   nelss 25917
16.15.2  Additional theory of functions   fninfp 25920
16.15.3  Extensions beyond function theory   gsumvsmul 25930
16.15.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 25939
16.15.6  Algebraic closure systems   cnacs 25943
16.15.7  Miscellanea 1. Map utilities   constmap 25954
16.15.8  Miscellanea for polynomials   ofmpteq 25963
16.15.9  Multivariate polynomials over the integers   cmzpcl 25965
16.15.10  Miscellanea for Diophantine sets 1   coeq0 25997
16.15.11  Diophantine sets 1: definitions   cdioph 26000
16.15.12  Diophantine sets 2 miscellanea   ellz1 26012
16.15.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26018
16.15.14  Diophantine sets 3: construction   diophrex 26021
16.15.15  Diophantine sets 4 miscellanea   2sbcrex 26030
16.15.16  Diophantine sets 4: Quantification   rexrabdioph 26041
16.15.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26048
16.15.18  Diophantine sets 6 miscellanea   fz1ssnn 26058
16.15.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26060
16.15.20  Pigeonhole Principle and cardinality helpers   fphpd 26065
16.15.21  A non-closed set of reals is infinite   rencldnfilem 26069
16.15.22  Miscellanea for Lagrange's theorem   icodiamlt 26071
16.15.23  Lagrange's rational approximation theorem   irrapxlem1 26073
16.15.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26080
16.15.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26087
16.15.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26129
16.15.27  Logarithm laws generalized to an arbitrary base   reglogcl 26141
16.15.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26149
16.15.29  X and Y sequences 1: Definition and recurrence laws   crmx 26151
16.15.30  Ordering and induction lemmas for the integers   monotuz 26192
16.15.31  X and Y sequences 2: Order properties   rmxypos 26200
16.15.32  Congruential equations   congtr 26218
16.15.33  Alternating congruential equations   acongid 26228
16.15.34  Additional theorems on integer divisibility   bezoutr 26238
16.15.35  X and Y sequences 3: Divisibility properties   jm2.18 26247
16.15.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26264
16.15.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26274
16.15.38  Uncategorized stuff not associated with a major project   setindtr 26283
16.15.39  More equivalents of the Axiom of Choice   axac10 26292
16.15.40  Finitely generated left modules   clfig 26331
16.15.41  Noetherian left modules I   clnm 26339
16.15.42  Addenda for structure powers   pwssplit0 26353
16.15.43  Direct sum of left modules   cdsmm 26363
16.15.44  Free modules   cfrlm 26378
16.15.45  Every set admits a group structure iff choice   unxpwdom3 26422
16.15.46  Independent sets and families   clindf 26440
16.15.47  Characterization of free modules   lmimlbs 26472
16.15.48  Noetherian rings and left modules II   clnr 26479
16.15.49  Hilbert's Basis Theorem   cldgis 26491
16.15.50  Additional material on polynomials [DEPRECATED]   cmnc 26501
16.15.51  Degree and minimal polynomial of algebraic numbers   cdgraa 26511
16.15.52  Algebraic integers I   citgo 26528
16.15.53  Finite cardinality [SO]   en1uniel 26546
16.15.54  Words in monoids and ordered group sum   issubmd 26549
16.15.55  Transpositions in the symmetric group   cpmtr 26550
16.15.56  The sign of a permutation   cpsgn 26580
16.15.57  The matrix algebra   cmmul 26605
16.15.58  The determinant   cmdat 26649
16.15.59  Endomorphism algebra   cmend 26655
16.15.60  Subfields   csdrg 26669
16.15.61  Cyclic groups and order   idomrootle 26677
16.15.62  Cyclotomic polynomials   ccytp 26687
16.15.63  Miscellaneous topology   fgraphopab 26695
16.16  Mathbox for Steve Rodriguez
16.16.1  Miscellanea   iso0 26702
16.16.2  Function operations   caofcan 26706
16.16.3  Calculus   lhe4.4ex1a 26712
16.17  Mathbox for Andrew Salmon
16.17.1  Principia Mathematica * 10   pm10.12 26719
16.17.2  Principia Mathematica * 11   2alanimi 26733
16.17.3  Predicate Calculus   sbeqal1 26763
16.17.4  Principia Mathematica * 13 and * 14   pm13.13a 26774
16.17.5  Set Theory   elnev 26805
16.17.7  Geometry   cplusr 26829
16.18  Mathbox for Jarvin Udandy
16.19  Mathbox for David A. Wheeler
16.19.2  Greater than, greater than or equal to.   cge-real 26879
16.19.3  Hyperbolic trig functions   csinh 26889
16.19.4  Reciprocal trig functions (sec, csc, cot)   csec 26900
16.19.5  Identities for "if"   ifnmfalse 26922
16.19.6  Not-member-of   AnelBC 26923
16.19.7  Decimal point   cdp2 26924
16.19.8  Signum (sgn or sign) function   csgn 26932
16.19.9  Ceiling function   ccei 26942
16.19.10  Logarithm laws generalized to an arbitrary base   clogb 26946
16.19.11  Miscellaneous   2m1e1 26951
16.20  Mathbox for Alan Sare
16.20.1  Conventional Metamath proofs, some derived from VD proofs   iidn3 26955
16.20.2  What is Virtual Deduction?   wvd1 27030
16.20.3  Virtual Deduction Theorems   df-vd1 27031
16.20.4  Theorems proved using virtual deduction   trsspwALT 27282
16.20.5  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 27312
16.20.6  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 27379
16.20.7  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 27383
16.20.8  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 27390
16.20.9  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 27393
16.21  Mathbox for Jonathan Ben-Naim
16.21.1  First order logic and set theory   bnj170 27412
16.21.2  Well founded induction and recursion   bnj110 27579
16.21.3  The existence of a minimal element in certain classes   bnj69 27729
16.21.4  Well-founded induction   bnj1204 27731
16.21.5  Well-founded recursion, part 1 of 3   bnj60 27781
16.21.6  Well-founded recursion, part 2 of 3   bnj1500 27787
16.21.7  Well-founded recursion, part 3 of 3   bnj1522 27791
16.22  Mathbox for Norm Megill
16.22.1  Obsolete experiments to study ax-12o   ax12-2 27792
16.22.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 27853
16.22.4  Functionals and kernels of a left vector space (or module)   clfn 27936
16.22.5  Opposite rings and dual vector spaces   cld 28002
16.22.6  Ortholattices and orthomodular lattices   cops 28051
16.22.7  Atomic lattices with covering property   ccvr 28141
16.22.8  Hilbert lattices   chlt 28229
16.22.9  Projective geometries based on Hilbert lattices   clln 28369
16.22.10  Construction of a vector space from a Hilbert lattice   cdlema1N 28669
16.22.11  Construction of involution and inner product from a Hilbert lattice   clpoN 30359

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