PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
1.1  Pre-logic
1.2  Propositional calculus
1.3  Other axiomatizations related to classical propositional calculus
1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
1.6  Existential uniqueness
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 Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
5.1  Construction and axiomatization of real and complex numbers
5.2  Derive the basic properties from the field axioms
5.3  Real and complex numbers - basic operations
5.4  Integer sets
5.5  Order sets
5.6  Elementary integer functions
5.7  Words over a set
5.8  Reflexive and transitive closures of relations
5.9  Elementary real and complex functions
5.10  Elementary limits and convergence
5.11  Elementary trigonometry
5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
6.1  Elementary properties of divisibility
6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
7.1  Extensible structures
7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
8.1  Categories
8.2  Arrows (disjointified hom-sets)
8.3  Examples of categories
8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
9.1  Presets and directed sets using extensible structures
9.2  Posets and lattices using extensible structures
PART 10  BASIC ALGEBRAIC STRUCTURES
10.1  Monoids
10.2  Groups
10.3  Abelian groups
10.4  Rings
10.5  Division rings and fields
10.6  Left modules
10.7  Vector spaces
10.8  Ideals
10.9  Associative algebras
10.10  Abstract multivariate polynomials
10.11  The complex numbers as an algebraic extensible structure
10.12  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
11.1  Vectors and free modules
11.2  Matrices
11.3  The determinant
11.4  Polynomial matrices
11.5  The characteristic polynomial
PART 12  BASIC TOPOLOGY
12.1  Topology
12.2  Filters and filter bases
12.3  Uniform Structures and Spaces
12.4  Metric spaces
12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
13.1  Continuity
13.2  Integrals
13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
14.1  Polynomials
14.2  Sequences and series
14.3  Basic trigonometry
14.4  Basic number theory
PART 15  ELEMENTARY GEOMETRY
15.1  Definition and Tarski's Axioms of Geometry
15.2  Tarskian Geometry
15.3  Properties of geometries
15.4  Geometry in Hilbert spaces
PART 16  GRAPH THEORY
16.1  Vertices and edges
16.2  Undirected graphs
16.3  Walks, paths and cycles
16.4  Eulerian paths and the Konigsberg Bridge problem
16.5  The Friendship Theorem
PART 17  GUIDES AND MISCELLANEA
17.1  Guides (conventions, explanations, and examples)
17.2  Humor
17.3  (Future - to be reviewed and classified)
PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
18.1  Additional material on group theory (deprecated)
18.2  Complex vector spaces
18.3  Normed complex vector spaces
18.4  Operators on complex vector spaces
18.5  Inner product (pre-Hilbert) spaces
18.6  Complex Banach spaces
18.7  Complex Hilbert spaces
PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
19.1  Axiomatization of complex pre-Hilbert spaces
19.2  Inner product and norms
19.3  Cauchy sequences and completeness axiom
19.4  Subspaces and projections
19.5  Properties of Hilbert subspaces
19.6  Operators on Hilbert spaces
19.7  States on a Hilbert lattice and Godowski's equation
19.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 20  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
20.1  Mathboxes for user contributions
20.2  Mathbox for Stefan Allan
20.3  Mathbox for Thierry Arnoux
20.4  Mathbox for Jonathan Ben-Naim
20.5  Mathbox for Mario Carneiro
20.6  Mathbox for Filip Cernatescu
20.7  Mathbox for Paul Chapman
20.8  Mathbox for Scott Fenton
20.9  Mathbox for Jeff Hankins
20.10  Mathbox for Anthony Hart
20.11  Mathbox for Chen-Pang He
20.12  Mathbox for Jeff Hoffman
20.13  Mathbox for Asger C. Ipsen
20.14  Mathbox for BJ
20.15  Mathbox for Jim Kingdon
20.16  Mathbox for ML
20.17  Mathbox for Wolf Lammen
20.18  Mathbox for Brendan Leahy
20.20  Mathbox for Giovanni Mascellani
20.21  Mathbox for Rodolfo Medina
20.22  Mathbox for Norm Megill
20.23  Mathbox for OpenAI
20.24  Mathbox for Stefan O'Rear
20.25  Mathbox for Jon Pennant
20.26  Mathbox for Richard Penner
20.27  Mathbox for Stanislas Polu
20.28  Mathbox for Steve Rodriguez
20.29  Mathbox for Andrew Salmon
20.30  Mathbox for Alan Sare
20.31  Mathbox for Glauco Siliprandi
20.32  Mathbox for Saveliy Skresanov
20.33  Mathbox for Jarvin Udandy
20.34  Mathbox for Alexander van der Vekens
20.35  Mathbox for Emmett Weisz
20.36  Mathbox for David A. Wheeler
20.37  Mathbox for Kunhao Zheng

(* means the section header has a description)
*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
*1.1  Pre-logic
*1.1.1  Inferences for assisting proof development   a1ii 1
*1.2  Propositional calculus
1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
*1.2.2  The axioms of propositional calculus   ax-mp 5
*1.2.3  Logical implication   mp2 9
*1.2.4  Logical negation   con4 112
*1.2.5  Logical equivalence   wb 196
*1.2.6  Logical disjunction and conjunction   wo 383
*1.2.7  Miscellaneous theorems of propositional calculus   pm5.62 957
*1.2.8  The conditional operator for propositions   wif 1011
*1.2.9  The weak deduction theorem   elimh 1029
1.2.10  Abbreviated conjunction and disjunction of three wff's   w3o 1035
1.2.11  Logical 'nand' (Sheffer stroke)   wnan 1444
1.2.12  Logical 'xor'   wxo 1461
1.2.13  True and false constants   wal 1478
*1.2.13.1  Universal quantifier for use by df-tru   wal 1478
*1.2.13.2  Equality predicate for use by df-tru   cv 1479
1.2.13.3  Define the true and false constants   wtru 1481
*1.2.14  Truth tables   truantru 1503
1.3  Other axiomatizations related to classical propositional calculus
*1.3.1  Minimal implicational calculus   minimp 1557
1.3.2  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1563
1.3.3  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1580
*1.3.4  Derive Nicod's axiom from the standard axioms   nic-dfim 1591
1.3.5  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1597
1.3.6  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1616
1.3.7  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1620
1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1635
1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1658
1.3.10  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1671
*1.3.11  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1690
*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
*1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1701
1.4.1.1  Existential quantifier   wex 1701
1.4.1.2  Non-freeness predicate   wnf 1705
1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1719
1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1734
1.4.4  Axiom scheme ax-5 (Distinctness) - first use of \$d   ax-5 1836
*1.4.5  Equality predicate (continued)   weq 1871
1.4.6  Define proper substitution   wsb 1877
1.4.7  Axiom scheme ax-6 (Existence)   ax-6 1885
1.4.8  Axiom scheme ax-7 (Equality)   ax-7 1932
1.4.9  Membership predicate   wcel 1987
1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 1989
1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 1996
*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2002
*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2016
1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2031
1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2044
1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2245
1.6  Existential uniqueness
1.7  Other axiomatizations related to classical predicate calculus
*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2562
*1.7.2  Intuitionistic logic   axia1 2586
*PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
2.1.1  Introduce the Axiom of Extensionality   ax-ext 2601
2.1.2  Class abstractions (a.k.a. class builders)   cab 2607
2.1.3  Class form not-free predicate   wnfc 2748
2.1.4  Negated equality and membership   wne 2790
2.1.4.1  Negated equality   wne 2790
2.1.4.2  Negated membership   wnel 2893
2.1.5  Restricted quantification   wral 2908
2.1.6  The universal class   cvv 3190
*2.1.7  Conditional equality (experimental)   wcdeq 3405
2.1.9  Proper substitution of classes for sets   wsbc 3422
2.1.10  Proper substitution of classes for sets into classes   csb 3519
2.1.11  Define basic set operations and relations   cdif 3557
2.1.12  Subclasses and subsets   df-ss 3574
2.1.13  The difference, union, and intersection of two classes   difeq1 3705
2.1.13.1  The difference of two classes   difeq1 3705
2.1.13.2  The union of two classes   elun 3737
2.1.13.3  The intersection of two classes   elin 3780
2.1.13.4  The symmetric difference of two classes   csymdif 3827
2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 3838
2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 3876
2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 3889
2.1.14  The empty set   c0 3897
*2.1.15  "Weak deduction theorem" for set theory   cif 4064
2.1.16  Power classes   cpw 4136
2.1.17  Unordered and ordered pairs   snjust 4154
2.1.18  The union of a class   cuni 4409
2.1.19  The intersection of a class   cint 4447
2.1.20  Indexed union and intersection   ciun 4492
2.1.21  Disjointness   wdisj 4593
2.1.22  Binary relations   wbr 4623
2.1.23  Ordered-pair class abstractions (class builders)   copab 4682
2.1.24  Transitive classes   wtr 4722
2.2  ZF Set Theory - add the Axiom of Replacement
2.2.1  Introduce the Axiom of Replacement   ax-rep 4741
2.2.2  Derive the Axiom of Separation   axsep 4750
2.2.3  Derive the Null Set Axiom   zfnuleu 4756
2.2.4  Theorems requiring subset and intersection existence   nalset 4765
2.2.5  Theorems requiring empty set existence   class2set 4802
2.3  ZF Set Theory - add the Axiom of Power Sets
2.3.1  Introduce the Axiom of Power Sets   ax-pow 4813
2.3.2  Derive the Axiom of Pairing   zfpair 4875
2.3.3  Ordered pair theorem   opnz 4912
2.3.4  Ordered-pair class abstractions (cont.)   opabid 4952
2.3.5  Power class of union and intersection   pwin 4988
2.3.6  Epsilon and identity relations   cep 4993
2.3.7  Partial and complete ordering   wpo 5003
2.3.8  Founded and well-ordering relations   wfr 5040
2.3.9  Relations   cxp 5082
2.3.10  The Predecessor Class   cpred 5648
2.3.11  Well-founded induction   tz6.26 5680
2.3.12  Ordinals   word 5691
2.3.13  Definite description binder (inverted iota)   cio 5818
2.3.14  Functions   wfun 5851
2.3.15  Cantor's Theorem   canth 6573
2.3.16  Restricted iota (description binder)   crio 6575
2.3.17  Operations   co 6615
2.3.18  "Maps to" notation   mpt2ndm0 6840
2.3.19  Function operation   cof 6860
2.3.20  Proper subset relation   crpss 6901
2.4  ZF Set Theory - add the Axiom of Union
2.4.1  Introduce the Axiom of Union   ax-un 6914
2.4.2  Ordinals (continued)   ordon 6944
2.4.3  Transfinite induction   tfi 7015
2.4.4  The natural numbers (i.e. finite ordinals)   com 7027
2.4.5  Peano's postulates   peano1 7047
2.4.6  Finite induction (for finite ordinals)   find 7053
2.4.7  First and second members of an ordered pair   c1st 7126
*2.4.8  The support of functions   csupp 7255
*2.4.9  Special "Maps to" operations   opeliunxp2f 7296
2.4.10  Function transposition   ctpos 7311
2.4.11  Curry and uncurry   ccur 7351
2.4.12  Undefined values   cund 7358
2.4.13  Well-founded recursion   cwrecs 7366
2.4.14  Functions on ordinals; strictly monotone ordinal functions   iunon 7396
2.4.15  "Strong" transfinite recursion   crecs 7427
2.4.16  Recursive definition generator   crdg 7465
2.4.17  Finite recursion   frfnom 7490
2.4.18  Ordinal arithmetic   c1o 7513
2.4.19  Natural number arithmetic   nna0 7644
2.4.20  Equivalence relations and classes   wer 7699
2.4.21  The mapping operation   cmap 7817
2.4.22  Infinite Cartesian products   cixp 7868
2.4.23  Equinumerosity   cen 7912
2.4.24  Schroeder-Bernstein Theorem   sbthlem1 8030
2.4.25  Equinumerosity (cont.)   xpf1o 8082
2.4.26  Pigeonhole Principle   phplem1 8099
2.4.27  Finite sets   onomeneq 8110
2.4.28  Finitely supported functions   cfsupp 8235
2.4.29  Finite intersections   cfi 8276
2.4.30  Hall's marriage theorem   marypha1lem 8299
2.4.31  Supremum and infimum   csup 8306
2.4.32  Ordinal isomorphism, Hartog's theorem   coi 8374
2.4.33  Hartogs function, order types, weak dominance   char 8421
2.5  ZF Set Theory - add the Axiom of Regularity
2.5.1  Introduce the Axiom of Regularity   ax-reg 8457
2.5.2  Axiom of Infinity equivalents   inf0 8478
2.6  ZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity   ax-inf 8495
2.6.2  Existence of omega (the set of natural numbers)   omex 8500
2.6.3  Cantor normal form   ccnf 8518
2.6.4  Transitive closure   trcl 8564
2.6.5  Rank   cr1 8585
2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 8708
2.6.7  Cardinal numbers   ccrd 8721
2.6.8  Axiom of Choice equivalents   wac 8898
2.6.9  Cardinal number arithmetic   ccda 8949
2.6.10  The Ackermann bijection   ackbij2lem1 9001
2.6.11  Cofinality (without Axiom of Choice)   cflem 9028
2.6.12  Eight inequivalent definitions of finite set   sornom 9059
2.6.13  Hereditarily size-limited sets without Choice   itunifval 9198
*PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
3.1.1  Introduce the Axiom of Countable Choice   ax-cc 9217
3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9228
3.2  ZFC Set Theory - add the Axiom of Choice
3.2.1  Introduce the Axiom of Choice   ax-ac 9241
3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9276
3.2.3  Cardinal number theorems using Axiom of Choice   cardval 9328
3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 9356
3.2.5  Cofinality using Axiom of Choice   alephreg 9364
3.3  ZFC Axioms with no distinct variable requirements
3.4  The Generalized Continuum Hypothesis
3.4.1  Sets satisfying the Generalized Continuum Hypothesis   cgch 9402
3.4.2  Derivation of the Axiom of Choice   gchaclem 9460
*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
4.1  Inaccessibles
4.1.1  Weakly and strongly inaccessible cardinals   cwina 9464
4.1.2  Weak universes   cwun 9482
4.1.3  Tarski classes   ctsk 9530
4.1.4  Grothendieck universes   cgru 9572
4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 9605
4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 9608
4.2.3  Tarski map function   ctskm 9619
*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 9626
5.1.2  Final derivation of real and complex number postulates   axaddf 9926
5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9952
5.2  Derive the basic properties from the field axioms
5.2.1  Some deductions from the field axioms for complex numbers   cnex 9977
5.2.2  Infinity and the extended real number system   cpnf 10031
5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 10069
5.2.4  Ordering on reals   lttr 10074
5.2.5  Initial properties of the complex numbers   mul12 10162
5.3  Real and complex numbers - basic operations
5.3.2  Subtraction   cmin 10226
5.3.3  Multiplication   kcnktkm1cn 10421
5.3.4  Ordering on reals (cont.)   gt0ne0 10453
5.3.5  Reciprocals   ixi 10616
5.3.6  Division   cdiv 10644
5.3.7  Ordering on reals (cont.)   elimgt0 10819
5.3.8  Completeness Axiom and Suprema   fimaxre 10928
5.3.9  Imaginary and complex number properties   inelr 10970
5.3.10  Function operation analogue theorems   ofsubeq0 10977
5.4  Integer sets
5.4.1  Positive integers (as a subset of complex numbers)   cn 10980
5.4.2  Principle of mathematical induction   nnind 10998
*5.4.3  Decimal representation of numbers   c2 11030
*5.4.4  Some properties of specific numbers   neg1cn 11084
5.4.5  Simple number properties   halfcl 11217
5.4.6  The Archimedean property   nnunb 11248
5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11252
*5.4.8  Extended nonnegative integers   cxnn0 11323
5.4.9  Integers (as a subset of complex numbers)   cz 11337
5.4.10  Decimal arithmetic   cdc 11453
5.4.11  Upper sets of integers   cuz 11647
5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 11743
5.4.13  Rational numbers (as a subset of complex numbers)   cq 11748
5.4.14  Existence of the set of complex numbers   rpnnen1lem2 11774
5.5  Order sets
5.5.1  Positive reals (as a subset of complex numbers)   crp 11792
5.5.2  Infinity and the extended real number system (cont.)   cxne 11903
5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12094
5.5.4  Real number intervals   cioo 12133
5.5.5  Finite intervals of integers   cfz 12284
*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 12388
5.5.7  Half-open integer ranges   cfzo 12422
5.6  Elementary integer functions
5.6.1  The floor and ceiling functions   cfl 12547
5.6.2  The modulo (remainder) operation   cmo 12624
5.6.3  Miscellaneous theorems about integers   om2uz0i 12702
5.6.4  Strong induction over upper sets of integers   uzsinds 12742
5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 12745
5.6.6  The infinite sequence builder "seq" - extension   cseq 12757
5.6.7  Integer powers   cexp 12816
5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13010
5.6.9  Factorial function   cfa 13016
5.6.10  The binomial coefficient operation   cbc 13045
5.6.11  The ` # ` (set size) function   chash 13073
5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 13204
5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 13229
5.6.11.3  Finite induction on the size of the first component of a binary relation   brfi1indlem 13233
*5.7  Words over a set
5.7.1  Definitions and basic theorems   cword 13246
5.7.2  Last symbol of a word   lsw 13306
5.7.3  Concatenations of words   ccatfn 13312
5.7.4  Singleton words   ids1 13332
5.7.5  Concatenations with singleton words   ccatws1cl 13351
5.7.6  Subwords   swrdval 13371
5.7.7  Subwords of subwords   swrdswrdlem 13413
5.7.8  Subwords and concatenations   wrdcctswrd 13419
5.7.9  Subwords of concatenations   swrdccatfn 13435
5.7.10  Splicing words (substring replacement)   splval 13455
5.7.11  Reversing words   revval 13462
5.7.12  Repeated symbol words   reps 13470
*5.7.13  Cyclical shifts of words   ccsh 13487
5.7.14  Mapping words by a function   wrdco 13530
5.7.15  Longer string literals   cs2 13539
*5.8  Reflexive and transitive closures of relations
5.8.1  The reflexive and transitive properties of relations   coss12d 13661
5.8.2  Basic properties of closures   cleq1lem 13671
5.8.3  Definitions and basic properties of transitive closures   ctcl 13674
5.8.4  Exponentiation of relations   crelexp 13710
5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 13745
*5.8.6  Principle of transitive induction.   relexpindlem 13753
5.9  Elementary real and complex functions
5.9.1  The "shift" operation   cshi 13756
5.9.2  Signum (sgn or sign) function   csgn 13776
5.9.3  Real and imaginary parts; conjugate   ccj 13786
5.9.4  Square root; absolute value   csqrt 13923
5.10  Elementary limits and convergence
5.10.1  Superior limit (lim sup)   clsp 14151
5.10.2  Limits   cli 14165
5.10.3  Finite and infinite sums   csu 14366
5.10.4  The binomial theorem   binomlem 14505
5.10.5  The inclusion/exclusion principle   incexclem 14512
5.10.6  Infinite sums (cont.)   isumshft 14515
5.10.7  Miscellaneous converging and diverging sequences   divrcnv 14528
5.10.8  Arithmetic series   arisum 14536
5.10.9  Geometric series   expcnv 14540
5.10.10  Ratio test for infinite series convergence   cvgrat 14559
5.10.11  Mertens' theorem   mertenslem1 14560
5.10.12  Finite and infinite products   prodf 14563
5.10.12.1  Product sequences   prodf 14563
5.10.12.2  Non-trivial convergence   ntrivcvg 14573
5.10.12.3  Complex products   cprod 14579
5.10.12.4  Finite products   fprod 14615
5.10.12.5  Infinite products   iprodclim 14673
5.10.13  Falling and Rising Factorial   cfallfac 14679
5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 14721
5.11  Elementary trigonometry
5.11.1  The exponential, sine, and cosine functions   ce 14736
5.11.2  _e is irrational   eirrlem 14876
5.12  Cardinality of real and complex number subsets
5.12.1  Countability of integers and rationals   xpnnen 14883
5.12.2  The reals are uncountable   rpnnen2lem1 14887
*PART 6  ELEMENTARY NUMBER THEORY
6.1  Elementary properties of divisibility
6.1.1  Irrationality of square root of 2   sqr2irrlem 14921
6.1.2  Some Number sets are chains of proper subsets   nthruc 14924
6.1.3  The divides relation   cdvds 14926
*6.1.4  Even and odd numbers   evenelz 15003
6.1.5  The division algorithm   divalglem0 15059
6.1.6  Bit sequences   cbits 15084
6.1.7  The greatest common divisor operator   cgcd 15159
6.1.8  Bézout's identity   bezoutlem1 15199
6.1.9  Algorithms   nn0seqcvgd 15226
6.1.10  Euclid's Algorithm   eucalgval2 15237
*6.1.11  The least common multiple   clcm 15244
*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 15305
6.1.13  Cancellability of congruences   congr 15321
6.2  Elementary prime number theory
*6.2.1  Elementary properties   cprime 15328
*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 15366
6.2.3  Properties of the canonical representation of a rational   cnumer 15384
6.2.4  Euler's theorem   codz 15411
6.2.5  Arithmetic modulo a prime number   modprm1div 15445
6.2.6  Pythagorean Triples   coprimeprodsq 15456
6.2.7  The prime count function   cpc 15484
6.2.8  Pocklington's theorem   prmpwdvds 15551
6.2.9  Infinite primes theorem   unbenlem 15555
6.2.10  Sum of prime reciprocals   prmreclem1 15563
6.2.11  Fundamental theorem of arithmetic   1arithlem1 15570
6.2.12  Lagrange's four-square theorem   cgz 15576
6.2.13  Van der Waerden's theorem   cvdwa 15612
6.2.14  Ramsey's theorem   cram 15646
*6.2.15  Primorial function   cprmo 15678
*6.2.16  Prime gaps   prmgaplem1 15696
6.2.17  Decimal arithmetic (cont.)   dec2dvds 15710
6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 15743
6.2.19  Specific prime numbers   prmlem0 15755
6.2.20  Very large primes   1259lem1 15781
PART 7  BASIC STRUCTURES
7.1  Extensible structures
*7.1.1  Basic definitions   cstr 15796
7.1.2  Slot definitions   cplusg 15881
7.1.3  Definition of the structure product   crest 16021
7.1.4  Definition of the structure quotient   cordt 16099
7.2  Moore spaces
7.2.1  Moore closures   mrcflem 16206
7.2.2  Independent sets in a Moore system   mrisval 16230
7.2.3  Algebraic closure systems   isacs 16252
PART 8  BASIC CATEGORY THEORY
8.1  Categories
8.1.1  Categories   ccat 16265
8.1.2  Opposite category   coppc 16311
8.1.3  Monomorphisms and epimorphisms   cmon 16328
8.1.4  Sections, inverses, isomorphisms   csect 16344
*8.1.5  Isomorphic objects   ccic 16395
8.1.6  Subcategories   cssc 16407
8.1.7  Functors   cfunc 16454
8.1.8  Full & faithful functors   cful 16502
8.1.9  Natural transformations and the functor category   cnat 16541
8.1.10  Initial, terminal and zero objects of a category   cinito 16578
8.2  Arrows (disjointified hom-sets)
8.2.1  Identity and composition for arrows   cida 16643
8.3  Examples of categories
8.3.1  The category of sets   csetc 16665
8.3.2  The category of categories   ccatc 16684
*8.3.3  The category of extensible structures   fncnvimaeqv 16700
8.4  Categorical constructions
8.4.1  Product of categories   cxpc 16748
8.4.2  Functor evaluation   cevlf 16789
8.4.3  Hom functor   chof 16828
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 16880
9.2.2  Lattices   clat 16985
9.2.3  The dual of an ordered set   codu 17068
9.2.4  Subset order structures   cipo 17091
9.2.5  Distributive lattices   latmass 17128
9.2.6  Posets and lattices as relations   cps 17138
9.2.7  Directed sets, nets   cdir 17168
PART 10  BASIC ALGEBRAIC STRUCTURES
10.1  Monoids
*10.1.1  Magmas   cplusf 17179
*10.1.2  Identity elements   mgmidmo 17199
*10.1.3  Ordered sums in a magma   gsumvalx 17210
*10.1.4  Semigroups   csgrp 17223
*10.1.5  Definition and basic properties of monoids   cmnd 17234
10.1.6  Monoid homomorphisms and submonoids   cmhm 17273
*10.1.7  Ordered sums in a monoid   gsumvallem2 17312
10.1.8  Free monoids   cfrmd 17324
10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 17345
10.2  Groups
10.2.1  Definition and basic properties   cgrp 17362
*10.2.2  Group multiple operation   cmg 17480
10.2.3  Subgroups and Quotient groups   csubg 17528
10.2.4  Elementary theory of group homomorphisms   cghm 17597
10.2.5  Isomorphisms of groups   cgim 17639
10.2.6  Group actions   cga 17662
10.2.7  Centralizers and centers   ccntz 17688
10.2.8  The opposite group   coppg 17715
10.2.9  Symmetric groups   csymg 17737
*10.2.9.1  Definition and basic properties   csymg 17737
10.2.9.2  Cayley's theorem   cayleylem1 17772
10.2.9.3  Permutations fixing one element   symgfix2 17776
*10.2.9.4  Transpositions in the symmetric group   cpmtr 17801
10.2.9.5  The sign of a permutation   cpsgn 17849
10.2.10  p-Groups and Sylow groups; Sylow's theorems   cod 17884
10.2.11  Direct products   clsm 17989
10.2.12  Free groups   cefg 18059
10.3  Abelian groups
10.3.1  Definition and basic properties   ccmn 18133
10.3.2  Cyclic groups   ccyg 18219
10.3.3  Group sum operation   gsumval3a 18244
10.3.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 18319
10.3.5  Internal direct products   cdprd 18332
10.3.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 18404
10.4  Rings
10.4.1  Multiplicative Group   cmgp 18429
10.4.2  Ring unit   cur 18441
10.4.2.1  Semirings   csrg 18445
*10.4.2.2  The binomial theorem for semirings   srgbinomlem1 18480
10.4.3  Definition and basic properties of unital rings   crg 18487
10.4.4  Opposite ring   coppr 18562
10.4.5  Divisibility   cdsr 18578
10.4.6  Ring homomorphisms   crh 18652
10.5  Division rings and fields
10.5.1  Definition and basic properties   cdr 18687
10.5.2  Subrings of a ring   csubrg 18716
10.5.3  Absolute value (abstract algebra)   cabv 18756
10.5.4  Star rings   cstf 18783
10.6  Left modules
10.6.1  Definition and basic properties   clmod 18803
10.6.2  Subspaces and spans in a left module   clss 18872
10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 18959
10.6.4  Subspace sum; bases for a left module   clbs 19014
10.7  Vector spaces
10.7.1  Definition and basic properties   clvec 19042
10.8  Ideals
10.8.1  The subring algebra; ideals   csra 19108
10.8.2  Two-sided ideals and quotient rings   c2idl 19171
10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 19181
10.8.4  Nonzero rings and zero rings   cnzr 19197
10.8.5  Left regular elements. More kinds of rings   crlreg 19219
10.9  Associative algebras
10.9.1  Definition and basic properties   casa 19249
10.10  Abstract multivariate polynomials
10.10.1  Definition and basic properties   cmps 19291
10.10.2  Polynomial evaluation   ces 19444
*10.10.3  Additional definitions for (multivariate) polynomials   cmhp 19477
*10.10.4  Univariate polynomials   cps1 19485
10.10.5  Univariate polynomial evaluation   ces1 19618
10.11  The complex numbers as an algebraic extensible structure
10.11.1  Definition and basic properties   cpsmet 19670
*10.11.2  Ring of integers   zring 19758
10.11.3  Algebraic constructions based on the complex numbers   czrh 19788
10.11.4  Signs as subgroup of the complex numbers   cnmsgnsubg 19863
10.11.5  Embedding of permutation signs into a ring   zrhpsgnmhm 19870
10.11.6  The ordered field of real numbers   crefld 19890
10.12  Generalized pre-Hilbert and Hilbert spaces
10.12.1  Definition and basic properties   cphl 19909
10.12.2  Orthocomplements and closed subspaces   cocv 19944
10.12.3  Orthogonal projection and orthonormal bases   cpj 19984
*PART 11  BASIC LINEAR ALGEBRA
11.1  Vectors and free modules
*11.1.1  Direct sum of left modules   cdsmm 20015
*11.1.2  Free modules   cfrlm 20030
*11.1.3  Standard basis (unit vectors)   cuvc 20061
*11.1.4  Independent sets and families   clindf 20083
11.1.5  Characterization of free modules   lmimlbs 20115
*11.2  Matrices
*11.2.1  The matrix multiplication   cmmul 20129
*11.2.2  Square matrices   cmat 20153
*11.2.3  The matrix algebra   matmulr 20184
*11.2.4  Matrices of dimension 0 and 1   mat0dimbas0 20212
*11.2.5  The subalgebras of diagonal and scalar matrices   cdmat 20234
*11.2.6  Multiplication of a matrix with a "column vector"   cmvmul 20286
11.2.7  Replacement functions for a square matrix   cmarrep 20302
11.2.8  Submatrices   csubma 20322
11.3  The determinant
11.3.1  Definition and basic properties   cmdat 20330
11.3.2  Determinants of 2 x 2 -matrices   m2detleiblem1 20370
*11.3.4  Laplace expansion of determinants (special case)   symgmatr01lem 20399
11.3.5  Inverse matrix   invrvald 20422
*11.3.6  Cramer's rule   slesolvec 20425
*11.4  Polynomial matrices
11.4.1  Basic properties   pmatring 20438
*11.4.2  Constant polynomial matrices   ccpmat 20448
*11.4.3  Collecting coefficients of polynomial matrices   cdecpmat 20507
*11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 20537
*11.5  The characteristic polynomial
*11.5.1  Definition and basic properties   cchpmat 20571
*11.5.2  The characteristic factor function G   fvmptnn04if 20594
*11.5.3  The Cayley-Hamilton theorem   cpmadurid 20612
PART 12  BASIC TOPOLOGY
12.1  Topology
*12.1.1  Topological spaces   ctop 20638
12.1.1.1  Topologies   ctop 20638
12.1.1.2  Topologies on sets   ctopon 20655
12.1.1.3  Topological spaces   ctps 20676
12.1.2  Topological bases   ctb 20689
12.1.3  Examples of topologies   distop 20739
12.1.4  Closure and interior   ccld 20760
12.1.5  Neighborhoods   cnei 20841
12.1.6  Limit points and perfect sets   clp 20878
12.1.7  Subspace topologies   restrcl 20901
12.1.8  Order topology   ordtbaslem 20932
12.1.9  Limits and continuity in topological spaces   ccn 20968
12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 21050
12.1.11  Compactness   ccmp 21129
12.1.12  Bolzano-Weierstrass theorem   bwth 21153
12.1.13  Connectedness   cconn 21154
12.1.14  First- and second-countability   c1stc 21180
12.1.15  Local topological properties   clly 21207
12.1.16  Refinements   cref 21245
12.1.17  Compactly generated spaces   ckgen 21276
12.1.18  Product topologies   ctx 21303
12.1.19  Continuous function-builders   cnmptid 21404
12.1.20  Quotient maps and quotient topology   ckq 21436
12.1.21  Homeomorphisms   chmeo 21496
12.2  Filters and filter bases
12.2.1  Filter bases   elmptrab 21570
12.2.2  Filters   cfil 21589
12.2.3  Ultrafilters   cufil 21643
12.2.4  Filter limits   cfm 21677
12.2.5  Extension by continuity   ccnext 21803
12.2.6  Topological groups   ctmd 21814
12.2.7  Infinite group sum on topological groups   ctsu 21869
12.2.8  Topological rings, fields, vector spaces   ctrg 21899
12.3  Uniform Structures and Spaces
12.3.1  Uniform structures   cust 21943
12.3.2  The topology induced by an uniform structure   cutop 21974
12.3.3  Uniform Spaces   cuss 21997
12.3.4  Uniform continuity   cucn 22019
12.3.5  Cauchy filters in uniform spaces   ccfilu 22030
12.3.6  Complete uniform spaces   ccusp 22041
12.4  Metric spaces
12.4.1  Pseudometric spaces   ispsmet 22049
12.4.2  Basic metric space properties   cxme 22062
12.4.3  Metric space balls   blfvalps 22128
12.4.4  Open sets of a metric space   mopnval 22183
12.4.5  Continuity in metric spaces   metcnp3 22285
12.4.6  The uniform structure generated by a metric   metuval 22294
12.4.7  Examples of metric spaces   dscmet 22317
*12.4.8  Normed algebraic structures   cnm 22321
12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 22449
12.4.10  Topology on the reals   qtopbaslem 22502
12.4.11  Topological definitions using the reals   cii 22618
12.4.12  Path homotopy   chtpy 22706
12.4.13  The fundamental group   cpco 22740
12.5  Metric subcomplex vector spaces
12.5.1  Subcomplex modules   cclm 22802
*12.5.2  Subcomplex vector spaces   ccvs 22863
*12.5.3  Normed subcomplex vector spaces   isncvsngp 22889
12.5.4  Subcomplex pre-Hilbert space   ccph 22906
12.5.5  Convergence and completeness   ccfil 22990
12.5.6  Baire's Category Theorem   bcthlem1 23061
12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 23069
12.5.7.1  The complete ordered field of the real numbers   retopn 23107
12.5.8  Euclidean spaces   crrx 23111
12.5.9  Minimizing Vector Theorem   minveclem1 23135
12.5.10  Projection Theorem   pjthlem1 23148
PART 13  BASIC REAL AND COMPLEX ANALYSIS
13.1  Continuity
13.1.1  Intermediate value theorem   pmltpclem1 23157
13.2  Integrals
13.2.1  Lebesgue measure   covol 23171
13.2.2  Lebesgue integration   cmbf 23323
13.2.2.1  Lesbesgue integral   cmbf 23323
13.2.2.2  Lesbesgue directed integral   cdit 23550
13.3  Derivatives
13.3.1  Real and complex differentiation   climc 23566
13.3.1.1  Derivatives of functions of one complex or real variable   climc 23566
13.3.1.2  Results on real differentiation   dvferm1lem 23685
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
14.1  Polynomials
14.1.1  Polynomial degrees   cmdg 23751
14.1.2  The division algorithm for univariate polynomials   cmn1 23823
14.1.3  Elementary properties of complex polynomials   cply 23878
14.1.4  The division algorithm for polynomials   cquot 23983
14.1.5  Algebraic numbers   caa 24007
14.1.6  Liouville's approximation theorem   aalioulem1 24025
14.2  Sequences and series
14.2.1  Taylor polynomials and Taylor's theorem   ctayl 24045
14.2.2  Uniform convergence   culm 24068
14.2.3  Power series   pserval 24102
14.3  Basic trigonometry
14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 24135
14.3.2  Properties of pi = 3.14159...   pilem1 24143
14.3.3  Mapping of the exponential function   efgh 24225
14.3.4  The natural logarithm on complex numbers   clog 24239
*14.3.5  Logarithms to an arbitrary base   clogb 24436
14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 24465
14.3.8  Inverse trigonometric functions   casin 24523
14.3.9  The Birthday Problem   log2ublem1 24607
14.3.10  Areas in R^2   carea 24616
14.3.11  More miscellaneous converging sequences   rlimcnp 24626
14.3.12  Inequality of arithmetic and geometric means   cvxcl 24645
14.3.13  Euler-Mascheroni constant   cem 24652
14.3.14  Zeta function   czeta 24673
14.3.15  Gamma function   clgam 24676
14.4  Basic number theory
14.4.1  Wilson's theorem   wilthlem1 24728
14.4.2  The Fundamental Theorem of Algebra   ftalem1 24733
14.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 24741
14.4.4  Number-theoretical functions   ccht 24751
14.4.5  Perfect Number Theorem   mersenne 24886
14.4.6  Characters of Z/nZ   cdchr 24891
14.4.7  Bertrand's postulate   bcctr 24934
*14.4.8  Quadratic residues and the Legendre symbol   clgs 24953
*14.4.9  Gauss' Lemma   gausslemma2dlem0a 25015
14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 25076
14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 25092
14.4.13  The Prime Number Theorem   mudivsum 25153
14.4.14  Ostrowski's theorem   abvcxp 25238
*PART 15  ELEMENTARY GEOMETRY
15.1  Definition and Tarski's Axioms of Geometry
15.2  Tarskian Geometry
15.2.1  Congruence   tgcgrcomimp 25306
15.2.2  Betweenness   tgbtwntriv2 25316
15.2.3  Dimension   tglowdim1 25329
15.2.4  Betweenness and Congruence   tgifscgr 25337
15.2.5  Congruence of a series of points   ccgrg 25339
15.2.6  Motions   cismt 25361
15.2.7  Colinearity   tglng 25375
15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 25401
15.2.9  Less-than relation in geometric congruences   cleg 25411
15.2.10  Rays   chlg 25429
15.2.11  Lines   btwnlng1 25448
15.2.12  Point inversions   cmir 25481
15.2.13  Right angles   crag 25522
15.2.14  Half-planes   islnopp 25565
15.2.15  Midpoints and Line Mirroring   cmid 25598
15.2.16  Congruence of angles   ccgra 25633
15.2.17  Angle Comparisons   cinag 25660
15.2.18  Congruence Theorems   tgsas1 25669
15.2.19  Equilateral triangles   ceqlg 25679
15.3  Properties of geometries
15.3.1  Isomorphisms between geometries   f1otrgds 25683
15.4  Geometry in Hilbert spaces
15.4.1  Geometry in the complex plane   cchhllem 25701
15.4.2  Geometry in Euclidean spaces   cee 25702
15.4.2.1  Definition of the Euclidean space   cee 25702
15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 25727
15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 25791
*PART 16  GRAPH THEORY
*16.1  Vertices and edges
16.1.1  The edge function extractor for extensible structures   cedgf 25801
*16.1.2  Vertices and indexed edges   cvtx 25808
16.1.2.1  Definitions and basic properties   cvtx 25808
16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 25817
16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 25825
16.1.2.4  Representations of graphs without edges   snstrvtxval 25863
16.1.2.5  Degenerated cases of representations of graphs   vtxval0 25865
16.1.3  Edges as range of the edge function   cedg 25873
*16.2  Undirected graphs
16.2.1  Undirected hypergraphs   cuhgr 25881
16.2.2  Undirected pseudographs and multigraphs   cupgr 25905
*16.2.3  Loop-free graphs   umgrislfupgrlem 25946
16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 25950
*16.2.5  Undirected simple graphs   cuspgr 25970
16.2.6  Examples for graphs   usgr0e 26055
16.2.7  Subgraphs   csubgr 26086
16.2.8  Finite undirected simple graphs   cfusgr 26130
16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 26145
16.2.9.1  Definitions   cnbgr 26145
16.2.9.2  Neighbors   nbgrcl 26154
16.2.9.3  Universal vertices   uvtxaval 26208
16.2.9.4  Complete graphs   iscplgr 26231
16.2.10  Vertex degree   cvtxdg 26282
*16.2.11  Regular graphs   crgr 26355
16.3  Walks, paths and cycles
*16.3.1  Walks   cewlks 26395
16.3.2  Walks for loop-free graphs   lfgrwlkprop 26487
16.3.3  Trails   ctrls 26490
16.3.4  Paths and simple paths   cpths 26511
16.3.5  Closed walks   cclwlks 26569
16.3.6  Circuits and cycles   ccrcts 26582
*16.3.7  Walks as words   cwwlks 26620
16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 26724
16.3.9  Walks in regular graphs   rusgrnumwwlkl1 26764
*16.3.10  Closed walks as words   cclwwlks 26776
16.3.11  Examples for walks, trails and paths   0ewlk 26875
16.3.12  Connected graphs   cconngr 26946
16.4  Eulerian paths and the Konigsberg Bridge problem
*16.4.1  Eulerian paths   ceupth 26957
*16.4.2  The Königsberg Bridge problem   konigsbergvtx 27006
16.5  The Friendship Theorem
16.5.1  Friendship graphs - basics   cfrgr 27020
16.5.2  The friendship theorem for small graphs   frgr1v 27033
16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 27044
*16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 27061
PART 17  GUIDES AND MISCELLANEA
17.1  Guides (conventions, explanations, and examples)
*17.1.1  Conventions   conventions 27146
17.1.2  Natural deduction   natded 27148
*17.1.3  Natural deduction examples   ex-natded5.2 27149
17.1.4  Definitional examples   ex-or 27166
17.1.5  Other examples   aevdemo 27205
17.2  Humor
17.2.1  April Fool's theorem   avril1 27207
17.3  (Future - to be reviewed and classified)
17.3.1  Planar incidence geometry   cplig 27214
17.3.2  Algebra preliminaries   crpm 27227
*PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
*18.1  Additional material on group theory (deprecated)
18.1.1  Definitions and basic properties for groups   cgr 27231
18.1.2  Abelian groups   cablo 27286
18.2  Complex vector spaces
18.2.1  Definition and basic properties   cvc 27301
18.2.2  Examples of complex vector spaces   cnaddabloOLD 27324
18.3  Normed complex vector spaces
18.3.1  Definition and basic properties   cnv 27327
18.3.2  Examples of normed complex vector spaces   cnnv 27420
18.3.3  Induced metric of a normed complex vector space   imsval 27428
18.3.4  Inner product   cdip 27443
18.3.5  Subspaces   css 27464
18.4  Operators on complex vector spaces
18.4.1  Definitions and basic properties   clno 27483
18.5  Inner product (pre-Hilbert) spaces
18.5.1  Definition and basic properties   ccphlo 27555
18.5.2  Examples of pre-Hilbert spaces   cncph 27562
18.5.3  Properties of pre-Hilbert spaces   isph 27565
18.6  Complex Banach spaces
18.6.1  Definition and basic properties   ccbn 27606
18.6.2  Examples of complex Banach spaces   cnbn 27613
18.6.3  Uniform Boundedness Theorem   ubthlem1 27614
18.6.4  Minimizing Vector Theorem   minvecolem1 27618
18.7  Complex Hilbert spaces
18.7.1  Definition and basic properties   chlo 27629
18.7.2  Standard axioms for a complex Hilbert space   hlex 27642
18.7.3  Examples of complex Hilbert spaces   cnchl 27660
18.7.4  Subspaces   ssphl 27661
18.7.5  Hellinger-Toeplitz Theorem   htthlem 27662
*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
19.1  Axiomatization of complex pre-Hilbert spaces
19.1.1  Basic Hilbert space definitions   chil 27664
19.1.2  Preliminary ZFC lemmas   df-hnorm 27713
*19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 27726
*19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 27744
19.1.5  Vector operations   hvmulex 27756
19.1.6  Inner product postulates for a Hilbert space   ax-hfi 27824
19.2  Inner product and norms
19.2.1  Inner product   his5 27831
19.2.2  Norms   dfhnorm2 27867
19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 27905
19.3  Cauchy sequences and completeness axiom
19.3.1  Cauchy sequences and limits   hcau 27929
19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 27939
19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 27947
19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 27948
19.4  Subspaces and projections
19.4.1  Subspaces   df-sh 27952
19.4.2  Closed subspaces   df-ch 27966
19.4.3  Orthocomplements   df-oc 27997
19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 28055
19.4.5  Projection theorem   pjhthlem1 28138
19.4.6  Projectors   df-pjh 28142
19.5  Properties of Hilbert subspaces
19.5.1  Orthomodular law   omlsilem 28149
19.5.2  Projectors (cont.)   pjhtheu2 28163
19.5.3  Hilbert lattice operations   sh0le 28187
19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 28288
19.5.5  Commutes relation for Hilbert lattice elements   df-cm 28330
19.5.6  Foulis-Holland theorem   fh1 28365
19.5.7  Quantum Logic Explorer axioms   qlax1i 28374
19.5.8  Orthogonal subspaces   chscllem1 28384
19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 28401
19.5.10  Projectors (cont.)   pjorthi 28416
19.5.11  Mayet's equation E_3   mayete3i 28475
19.6  Operators on Hilbert spaces
*19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 28477
19.6.2  Zero and identity operators   df-h0op 28495
19.6.3  Operations on Hilbert space operators   hoaddcl 28505
19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 28586
19.6.5  Linear and continuous functionals and norms   df-nmfn 28592
19.6.7  Dirac bra-ket notation   df-bra 28597
19.6.8  Positive operators   df-leop 28599
19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 28600
19.6.10  Theorems about operators and functionals   nmopval 28603
19.6.11  Riesz lemma   riesz3i 28809
19.6.13  Quantum computation error bound theorem   unierri 28851
19.6.14  Dirac bra-ket notation (cont.)   branmfn 28852
19.6.15  Positive operators (cont.)   leopg 28869
19.6.16  Projectors as operators   pjhmopi 28893
19.7  States on a Hilbert lattice and Godowski's equation
19.7.1  States on a Hilbert lattice   df-st 28958
19.7.2  Godowski's equation   golem1 29018
19.8  Cover relation, atoms, exchange axiom, and modular symmetry
19.8.1  Covers relation; modular pairs   df-cv 29026
19.8.2  Atoms   df-at 29085
19.8.3  Superposition principle   superpos 29101
19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 29102
19.8.5  Irreducibility   chirredlem1 29137
19.8.6  Atoms (cont.)   atcvat3i 29143
19.8.7  Modular symmetry   mdsymlem1 29150
PART 20  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
20.1  Mathboxes for user contributions
20.1.1  Mathbox guidelines   mathbox 29189
20.2  Mathbox for Stefan Allan
20.3  Mathbox for Thierry Arnoux
20.3.1  Propositional Calculus - misc additions   bian1d 29194
20.3.2  Predicate Calculus   spc2ed 29200
20.3.2.1  Predicate Calculus - misc additions   spc2ed 29200
20.3.2.2  Restricted quantification - misc additions   ralcom4f 29204
20.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 29209
20.3.2.4  Existential "at most one" - misc additions   moel 29212
20.3.2.5  Existential uniqueness - misc additions   2reuswap2 29217
20.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 29221
20.3.3  General Set Theory   rabrab 29227
20.3.3.1  Class abstractions (a.k.a. class builders)   rabrab 29227
20.3.3.2  Image Sets   abrexdomjm 29233
20.3.3.3  Set relations and operations - misc additions   eqri 29239
20.3.3.4  Unordered pairs   elpreq 29248
20.3.3.5  Conditional operator - misc additions   ifeqeqx 29249
20.3.3.6  Set union   uniinn0 29253
20.3.3.7  Indexed union - misc additions   cbviunf 29259
20.3.3.8  Disjointness - misc additions   disjnf 29270
20.3.4  Relations and Functions   xpdisjres 29297
20.3.4.1  Relations - misc additions   xpdisjres 29297
20.3.4.2  Functions - misc additions   ac6sf2 29313
20.3.4.3  Operations - misc additions   mpt2mptxf 29361
20.3.4.4  Isomorphisms - misc. add.   gtiso 29362
20.3.4.5  Disjointness (additional proof requiring functions)   disjdsct 29364
20.3.4.6  First and second members of an ordered pair - misc additions   df1stres 29365
20.3.4.7  Supremum - misc additions   supssd 29371
20.3.4.8  Finite Sets   imafi2 29373
20.3.4.9  Countable Sets   snct 29375
20.3.5  Real and Complex Numbers   addeq0 29394
20.3.5.3  Extended reals - misc additions   xlemnf 29400
20.3.5.4  Real number intervals - misc additions   joiniooico 29421
20.3.5.5  Finite intervals of integers - misc additions   nndiffz1 29431
20.3.5.6  Half-open integer ranges - misc additions   iundisjfi 29438
20.3.5.7  The ` # ` (set size) function - misc additions   hashunif 29445
20.3.5.8  The greatest common divisor operator - misc. add   numdenneg 29446
20.3.5.9  Integers   nnindf 29448
20.3.5.10  Division in the extended real number system   cxdiv 29452
20.3.6  Prime Number Theory   bhmafibid1 29471
20.3.6.1  Fermat's two square theorem   bhmafibid1 29471
20.3.7  Extensible Structures   ressplusf 29477
20.3.7.1  Structure restriction operator   ressplusf 29477
20.3.7.2  The opposite group   oppgle 29480
20.3.7.3  Posets   ressprs 29482
20.3.7.4  Complete lattices   clatp0cl 29498
20.3.7.6  The extended nonnegative real numbers commutative monoid   xrge0base 29512
20.3.8  Algebra   abliso 29523
20.3.8.1  Monoids Homomorphisms   abliso 29523
20.3.8.2  Ordered monoids and groups   comnd 29524
20.3.8.3  Signum in an ordered monoid   csgns 29552
20.3.8.4  The Archimedean property for generic ordered algebraic structures   cinftm 29557
20.3.8.5  Semiring left modules   cslmd 29580
20.3.8.6  Finitely supported group sums - misc additions   gsumle 29606
20.3.8.7  Rings - misc additions   rngurd 29615
20.3.8.8  Ordered rings and fields   corng 29622
20.3.8.9  Ring homomorphisms - misc additions   rhmdvdsr 29645
20.3.8.10  Scalar restriction operation   cresv 29651
20.3.8.11  The commutative ring of gaussian integers   gzcrng 29666
20.3.8.12  The archimedean ordered field of real numbers   reofld 29667
20.3.9  Matrices   symgfcoeu 29672
20.3.9.1  The symmetric group   symgfcoeu 29672
20.3.9.2  Permutation Signs   psgndmfi 29673
20.3.9.3  Submatrices   csmat 29683
20.3.9.4  Matrix literals   clmat 29701
20.3.9.5  Laplace expansion of determinants   mdetpmtr1 29713
20.3.10  Topology   fvproj 29723
20.3.10.1  Open maps   fvproj 29723
20.3.10.2  Topology of the unit circle   qtopt1 29726
20.3.10.3  Refinements   reff 29730
20.3.10.4  Open cover refinement property   ccref 29733
20.3.10.5  Lindelöf spaces   cldlf 29743
20.3.10.6  Paracompact spaces   cpcmp 29746
20.3.10.7  Pseudometrics   cmetid 29753
20.3.10.8  Continuity - misc additions   hauseqcn 29765
20.3.10.9  Topology of the closed unit   unitsscn 29766
20.3.10.10  Topology of ` ( RR X. RR ) `   unicls 29773
20.3.10.11  Order topology - misc. additions   cnvordtrestixx 29783
20.3.10.12  Continuity in topological spaces - misc. additions   mndpluscn 29796
20.3.10.13  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 29802
20.3.10.14  Limits - misc additions   lmlim 29817
20.3.10.15  Univariate polynomials   pl1cn 29825
20.3.11  Uniform Stuctures and Spaces   chcmp 29826
20.3.11.1  Hausdorff uniform completion   chcmp 29826
20.3.12  Topology and algebraic structures   zringnm 29828
20.3.12.1  The norm on the ring of the integer numbers   zringnm 29828
20.3.12.2  Topological ` ZZ ` -modules   zlm0 29830
20.3.12.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 29840
20.3.12.4  Canonical embedding of the real numbers into a complete ordered field   crrh 29861
20.3.12.5  Embedding from the extended real numbers into a complete lattice   cxrh 29884
20.3.12.6  Canonical embeddings into the ordered field of the real numbers   zrhre 29887
*20.3.12.7  Topological Manifolds   cmntop 29890
20.3.13  Real and complex functions   nexple 29895
20.3.13.1  Integer powers - misc. additions   nexple 29895
20.3.13.2  Indicator Functions   cind 29896
20.3.13.3  Extended sum   cesum 29912
20.3.14  Mixed Function/Constant operation   cofc 29980
20.3.15  Abstract measure   csiga 29993
20.3.15.1  Sigma-Algebra   csiga 29993
20.3.15.2  Generated sigma-Algebra   csigagen 30024
*20.3.15.3  lambda and pi-Systems, Rings of Sets   ispisys 30038
20.3.15.4  The Borel algebra on the real numbers   cbrsiga 30067
20.3.15.5  Product Sigma-Algebra   csx 30074
20.3.15.6  Measures   cmeas 30081
20.3.15.7  The counting measure   cntmeas 30112
20.3.15.8  The Lebesgue measure - misc additions   voliune 30115
20.3.15.9  The Dirac delta measure   cdde 30118
20.3.15.10  The 'almost everywhere' relation   cae 30123
20.3.15.11  Measurable functions   cmbfm 30135
20.3.15.12  Borel Algebra on ` ( RR X. RR ) `   br2base 30154
*20.3.15.13  Caratheodory's extension theorem   coms 30176
20.3.16  Integration   itgeq12dv 30211
20.3.16.1  Lebesgue integral - misc additions   itgeq12dv 30211
20.3.16.2  Bochner integral   citgm 30212
20.3.17  Euler's partition theorem   oddpwdc 30239
20.3.18  Sequences defined by strong recursion   csseq 30268
20.3.19  Fibonacci Numbers   cfib 30281
20.3.20  Probability   cprb 30292
20.3.20.1  Probability Theory   cprb 30292
20.3.20.2  Conditional Probabilities   ccprob 30316
20.3.20.3  Real Valued Random Variables   crrv 30325
20.3.20.4  Preimage set mapping operator   corvc 30340
20.3.20.5  Distribution Functions   orvcelval 30353
20.3.20.6  Cumulative Distribution Functions   orvclteel 30357
20.3.20.7  Probabilities - example   coinfliplem 30363
20.3.20.8  Bertrand's Ballot Problem   ballotlemoex 30370
20.3.21  Signum (sgn or sign) function - misc. additions   sgncl 30423
20.3.22  Words over a set - misc additions   wrdres 30439
20.3.22.1  Operations on words   ccatmulgnn0dir 30441
20.3.23  Polynomials with real coefficients - misc additions   plymul02 30445
20.3.24  Descartes's rule of signs   signspval 30451
20.3.24.1  Sign changes in a word over real numbers   signspval 30451
20.3.24.2  Counting sign changes in a word over real numbers   signslema 30461
20.3.25  Number Theory   efcld 30491
20.3.26  Elementary Geometry   cstrkg2d 30502
*20.3.26.1  Two-dimension geometry   cstrkg2d 30502
20.3.26.2  Outer Five Segment (not used, no need to move to main)   cafs 30507
*20.4  Mathbox for Jonathan Ben-Naim
20.4.1  First-order logic and set theory   bnj170 30524
20.4.2  Well founded induction and recursion   bnj110 30689
20.4.3  The existence of a minimal element in certain classes   bnj69 30839
20.4.4  Well-founded induction   bnj1204 30841
20.4.5  Well-founded recursion, part 1 of 3   bnj60 30891
20.4.6  Well-founded recursion, part 2 of 3   bnj1500 30897
20.4.7  Well-founded recursion, part 3 of 3   bnj1522 30901
20.5  Mathbox for Mario Carneiro
20.5.1  Predicate calculus with all distinct variables   ax-7d 30902
20.5.2  Miscellaneous stuff   quartfull 30908
20.5.3  Derangements and the Subfactorial   deranglem 30909
20.5.4  The Erdős-Szekeres theorem   erdszelem1 30934
20.5.5  The Kuratowski closure-complement theorem   kur14lem1 30949
20.5.6  Retracts and sections   cretr 30960
20.5.7  Path-connected and simply connected spaces   cpconn 30962
20.5.8  Covering maps   ccvm 30998
20.5.9  Normal numbers   snmlff 31072
20.5.10  Godel-sets of formulas   cgoe 31076
20.5.11  Models of ZF   cgze 31104
*20.5.12  Metamath formal systems   cmcn 31118
20.5.13  Grammatical formal systems   cm0s 31243
20.5.14  Models of formal systems   cmuv 31261
20.5.15  Splitting fields   citr 31283
20.5.16  p-adic number fields   czr 31299
*20.6  Mathbox for Filip Cernatescu
20.7  Mathbox for Paul Chapman
20.7.1  Real and complex numbers (cont.)   climuzcnv 31326
20.7.2  Miscellaneous theorems   elfzm12 31330
20.8  Mathbox for Scott Fenton
20.8.1  ZFC Axioms in primitive form   axextprim 31339
20.8.2  Untangled classes   untelirr 31346
20.8.3  Extra propositional calculus theorems   3orel1 31353
20.8.4  Misc. Useful Theorems   nepss 31361
20.8.5  Properties of real and complex numbers   sqdivzi 31371
20.8.6  Infinite products   iprodefisumlem 31387
20.8.7  Factorial limits   faclimlem1 31390
20.8.8  Greatest common divisor and divisibility   pdivsq 31396
20.8.9  Properties of relationships   brtp 31400
20.8.10  Properties of functions and mappings   funpsstri 31420
20.8.11  Epsilon induction   setinds 31437
20.8.12  Ordinal numbers   elpotr 31440
20.8.13  Defined equality axioms   axextdfeq 31457
20.8.14  Hypothesis builders   hbntg 31465
20.8.15  (Trans)finite Recursion Theorems   tfisg 31470
20.8.16  Transitive closure under a relationship   ctrpred 31471
20.8.17  Founded Induction   frmin 31493
20.8.18  Ordering Ordinal Sequences   orderseqlem 31503
20.8.19  Well-founded zero, successor, and limits   cwsuc 31506
20.8.20  Founded Recursion   frr3g 31533
20.8.21  Surreal Numbers   csur 31547
20.8.22  Surreal Numbers: Ordering   sltsolem1 31581
20.8.23  Surreal Numbers: Birthday Function   bdayfo 31591
20.8.24  Surreal Numbers: Density   fvnobday 31598
20.8.25  Surreal Numbers: Upper and Lower Bounds   nobndlem1 31608
20.8.26  Surreal Numbers: Full-Eta Property   nosepnelem 31618
20.8.27  Quantifier-free definitions   ctxp 31631
20.8.28  Alternate ordered pairs   caltop 31758
20.8.29  Geometry in the Euclidean space   cofs 31784
20.8.29.1  Congruence properties   cofs 31784
20.8.29.2  Betweenness properties   btwntriv2 31814
20.8.29.3  Segment Transportation   ctransport 31831
20.8.29.4  Properties relating betweenness and congruence   cifs 31837
20.8.29.5  Connectivity of betweenness   btwnconn1lem1 31889
20.8.29.6  Segment less than or equal to   csegle 31908
20.8.29.7  Outside of relationship   coutsideof 31921
20.8.29.8  Lines and Rays   cline2 31936
20.8.30  Forward difference   cfwddif 31960
20.8.31  Rank theorems   rankung 31968
20.8.32  Hereditarily Finite Sets   chf 31974
20.9  Mathbox for Jeff Hankins
20.9.1  Miscellany   a1i14 31989
20.9.2  Basic topological facts   topbnd 32014
20.9.3  Topology of the real numbers   ivthALT 32025
20.9.4  Refinements   cfne 32026
20.9.5  Neighborhood bases determine topologies   neibastop1 32049
20.9.6  Lattice structure of topologies   topmtcl 32053
20.9.7  Filter bases   fgmin 32060
20.9.8  Directed sets, nets   tailfval 32062
20.10  Mathbox for Anthony Hart
20.10.1  Propositional Calculus   tb-ax1 32073
20.10.2  Predicate Calculus   allt 32095
20.10.3  Misc. Single Axiom Systems   meran1 32105
20.10.4  Connective Symmetry   negsym1 32111
20.11  Mathbox for Chen-Pang He
20.11.1  Ordinal topology   ontopbas 32122
20.12  Mathbox for Jeff Hoffman
20.12.1  Inferences for finite induction on generic function values   fveleq 32145
20.12.2  gdc.mm   nnssi2 32149
20.13  Mathbox for Asger C. Ipsen
20.13.1  Continuous nowhere differentiable functions   dnival 32156
*20.14  Mathbox for BJ
*20.14.1  Propositional calculus   bj-mp2c 32226
*20.14.1.1  Derived rules of inference   bj-mp2c 32226
*20.14.1.2  A syntactic theorem   bj-0 32228
20.14.1.3  Minimal implicational calculus   bj-a1k 32230
20.14.1.4  Positive calculus   bj-orim2 32236
20.14.1.5  Implication and negation   pm4.81ALT 32241
*20.14.1.6  Disjunction   bj-jaoi1 32251
*20.14.1.7  Logical equivalence   bj-dfbi4 32253
20.14.1.8  The conditional operator for propositions   bj-consensus 32257
*20.14.1.9  Propositional calculus: miscellaneous   bj-imbi12 32262
*20.14.2  Modal logic   bj-axdd2 32271
*20.14.3  Provability logic   cprvb 32277
*20.14.4  First-order logic   bj-genr 32286
20.14.4.4  Equality and substitution   bj-ssbjust 32313
20.14.4.7  Membership predicate, ax-8 and ax-9   bj-elequ2g 32361
*20.14.4.11  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 32412
*20.14.4.12  Strengthenings of theorems of the main part   bj-sb3b 32500
*20.14.4.13  Distinct var metavariables   bj-hbaeb2 32501
*20.14.4.14  Around ~ equsal   bj-equsal1t 32505
*20.14.4.15  Some Principia Mathematica proofs   stdpc5t 32510
20.14.4.16  Alternate definition of substitution   bj-sbsb 32520
20.14.4.17  Lemmas for substitution   bj-sbf3 32522
20.14.4.18  Existential uniqueness   bj-eu3f 32525
*20.14.4.19  First-logic: miscellaneous   bj-nfdiOLD 32527
20.14.5  Set theory   eliminable1 32538
*20.14.5.1  Eliminability of class terms   eliminable1 32538
*20.14.5.2  Classes without extensionality   bj-cleljustab 32545
*20.14.5.3  The class-form not-free predicate   bj-nfcsym 32586
*20.14.5.4  Proposal for the definitions of class membership and class equality   bj-ax8 32587
*20.14.5.5  Lemmas for class substitution   bj-sbeqALT 32595
20.14.5.6  Removing some dv conditions   bj-exlimmpi 32605
*20.14.5.7  Class abstractions   bj-unrab 32622
*20.14.5.8  Restricted non-freeness   wrnf 32630
*20.14.5.10  Some disjointness results   bj-n0i 32635
*20.14.5.11  Complements on direct products   bj-xpimasn 32642
*20.14.5.12  "Singletonization" and tagging   bj-sels 32650
*20.14.5.13  Tuples of classes   bj-cproj 32678
*20.14.5.14  Set theory: miscellaneous   bj-disj2r 32713
*20.14.5.15  Elementwise intersection (families of sets induced on a subset)   bj-rest00 32724
20.14.5.16  Topology (complements)   cmoo 32743
20.14.5.17  Maps-to notation for functions with three arguments   bj-0nelmpt 32745
*20.14.5.18  Currying   cfset 32751
*20.14.6  Extended real and complex numbers, real and complex projectives lines   bj-elid 32757
*20.14.6.1  Diagonal in a Cartesian square   bj-elid 32757
*20.14.6.2  Extended numbers and projective lines as sets   cinftyexpi 32765
*20.14.6.4  Argument, multiplication and inverse   cprcpal 32800
*20.14.7  Monoids   bj-cmnssmnd 32808
*20.14.7.1  Finite sums in monoids   cfinsum 32817
*20.14.8  Affine, Euclidean, and Cartesian geometry   crrvec 32820
*20.14.8.1  Convex hull in real vector spaces   crrvec 32820
*20.14.8.2  Complex numbers (supplements)   bj-subcom 32826
*20.14.8.3  Barycentric coordinates   bj-bary1lem 32832
20.15  Mathbox for Jim Kingdon
20.16  Mathbox for ML
20.17  Mathbox for Wolf Lammen
20.17.1  1. Bootstrapping   wl-section-boot 32915
20.17.2  Implication chains   wl-section-impchain 32939
20.17.3  An alternative axiom ~ ax-13   ax-wl-13v 32957
20.17.4  Other stuff   wl-jarri 32959
20.17.5  1. Bootstrapping classes   wcel-wl 33044
20.18  Mathbox for Brendan Leahy
20.19.1  Logic and set theory   anim12da 33177
20.19.2  Real and complex numbers; integers   filbcmb 33206
20.19.3  Sequences and sums   sdclem2 33209
20.19.4  Topology   subspopn 33219
20.19.5  Metric spaces   metf1o 33222
20.19.6  Continuous maps and homeomorphisms   constcncf 33229
20.19.7  Boundedness   ctotbnd 33236
20.19.8  Isometries   cismty 33268
20.19.9  Heine-Borel Theorem   heibor1lem 33279
20.19.10  Banach Fixed Point Theorem   bfplem1 33292
20.19.11  Euclidean space   crrn 33295
20.19.12  Intervals (continued)   ismrer1 33308
20.19.13  Operation properties   cass 33312
20.19.14  Groups and related structures   cmagm 33318
20.19.15  Group homomorphism and isomorphism   cghomOLD 33353
20.19.16  Rings   crngo 33364
20.19.17  Division Rings   cdrng 33418
20.19.18  Ring homomorphisms   crnghom 33430
20.19.19  Commutative rings   ccm2 33459
20.19.20  Ideals   cidl 33477
20.19.21  Prime rings and integral domains   cprrng 33516
20.19.22  Ideal generators   cigen 33529
20.20  Mathbox for Giovanni Mascellani
*20.20.1  Tools for automatic proof building   efald2 33548
*20.20.2  Tseitin axioms   fald 33607
*20.20.3  Equality deductions   iuneq2f 33634
*20.20.4  Miscellanea   scottexf 33647
20.21  Mathbox for Rodolfo Medina
20.21.1  Partitions   prtlem60 33653
*20.22  Mathbox for Norm Megill
*20.22.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 33687
*20.22.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 33697
*20.22.3  Legacy theorems using obsolete axioms   ax5ALT 33711
20.22.4  Experiments with weak deduction theorem   elimhyps 33766
20.22.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 33780
20.22.7  Functionals and kernels of a left vector space (or module)   clfn 33863
20.22.8  Opposite rings and dual vector spaces   cld 33929
20.22.9  Ortholattices and orthomodular lattices   cops 33978
20.22.10  Atomic lattices with covering property   ccvr 34068
20.22.11  Hilbert lattices   chlt 34156
20.22.12  Projective geometries based on Hilbert lattices   clln 34296
20.22.13  Construction of a vector space from a Hilbert lattice   cdlema1N 34596
20.22.14  Construction of involution and inner product from a Hilbert lattice   clpoN 36288
20.23  Mathbox for OpenAI
20.24  Mathbox for Stefan O'Rear
20.24.1  Additional elementary logic and set theory   moxfr 36774
20.24.2  Additional theory of functions   imaiinfv 36775
20.24.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 36780
20.24.5  Algebraic closure systems   cnacs 36784
20.24.6  Miscellanea 1. Map utilities   constmap 36795
20.24.7  Miscellanea for polynomials   mptfcl 36802
20.24.8  Multivariate polynomials over the integers   cmzpcl 36803
20.24.9  Miscellanea for Diophantine sets 1   coeq0i 36835
20.24.10  Diophantine sets 1: definitions   cdioph 36837
20.24.11  Diophantine sets 2 miscellanea   ellz1 36849
20.24.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 36855
20.24.13  Diophantine sets 3: construction   diophrex 36858
20.24.14  Diophantine sets 4 miscellanea   2sbcrex 36867
20.24.15  Diophantine sets 4: Quantification   rexrabdioph 36877
20.24.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 36884
20.24.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 36894
20.24.18  Pigeonhole Principle and cardinality helpers   fphpd 36899
20.24.19  A non-closed set of reals is infinite   rencldnfilem 36903
20.24.20  Lagrange's rational approximation theorem   irrapxlem1 36905
20.24.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 36912
20.24.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 36919
20.24.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 36961
*20.24.24  Logarithm laws generalized to an arbitrary base   reglogcl 36973
20.24.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 36981
20.24.26  X and Y sequences 1: Definition and recurrence laws   crmx 36983
20.24.27  Ordering and induction lemmas for the integers   monotuz 37025
20.24.28  X and Y sequences 2: Order properties   rmxypos 37033
20.24.29  Congruential equations   congtr 37051
20.24.30  Alternating congruential equations   acongid 37061
20.24.31  Additional theorems on integer divisibility   coprmdvdsb 37071
20.24.32  X and Y sequences 3: Divisibility properties   jm2.18 37074
20.24.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 37091
20.24.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 37101
20.24.35  Uncategorized stuff not associated with a major project   setindtr 37110
20.24.36  More equivalents of the Axiom of Choice   axac10 37119
20.24.37  Finitely generated left modules   clfig 37156
20.24.38  Noetherian left modules I   clnm 37164
20.24.39  Addenda for structure powers   pwssplit4 37178
20.24.40  Every set admits a group structure iff choice   unxpwdom3 37184
20.24.41  Noetherian rings and left modules II   clnr 37199
20.24.42  Hilbert's Basis Theorem   cldgis 37211
20.24.43  Additional material on polynomials [DEPRECATED]   cmnc 37221
20.24.44  Degree and minimal polynomial of algebraic numbers   cdgraa 37230
20.24.45  Algebraic integers I   citgo 37247
20.24.46  Endomorphism algebra   cmend 37265
20.24.47  Subfields   csdrg 37285
20.24.48  Cyclic groups and order   idomrootle 37293
20.24.49  Cyclotomic polynomials   ccytp 37300
20.24.50  Miscellaneous topology   fgraphopab 37308
20.25  Mathbox for Jon Pennant
20.26  Mathbox for Richard Penner
20.26.1  Short Studies   ifpan123g 37323
20.26.1.1  Additional work on conditional logical operator   ifpan123g 37323
20.26.1.2  Sophisms   rp-fakeimass 37377
*20.26.1.3  Finite Sets   rp-isfinite5 37383
20.26.1.4  Infinite Sets   pwelg 37385
*20.26.1.5  Finite intersection property   fipjust 37390
20.26.1.6  RP ADDTO: Subclasses and subsets   rababg 37399
20.26.1.7  RP ADDTO: The intersection of a class   elintabg 37400
20.26.1.8  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 37403
20.26.1.9  RP ADDTO: Relations   xpinintabd 37406
*20.26.1.10  RP ADDTO: Functions   elmapintab 37422
*20.26.1.11  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 37426
20.26.1.12  RP ADDTO: First and second members of an ordered pair   elcnvlem 37427
20.26.1.13  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 37430
20.26.1.14  RP ADDTO: Basic properties of closures   cleq2lem 37434
20.26.1.15  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 37457
20.26.2  Additional statements on relations and subclasses   al3im 37458
20.26.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 37477
20.26.2.2  Reflexive closures   crcl 37484
*20.26.2.3  Finite relationship composition.   relexp2 37489
20.26.2.4  Transitive closure of a relation   dftrcl3 37532
*20.26.2.5  Adapted from Frege   frege77d 37558
*20.26.3  Propositions from _Begriffsschrift_   dfxor4 37578
*20.26.3.1  _Begriffsschrift_ Chapter I   dfxor4 37578
*20.26.3.2  _Begriffsschrift_ Notation hints   rp-imass 37586
20.26.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 37605
20.26.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 37644
*20.26.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 37671
20.26.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 37702
20.26.3.7  _Begriffsschrift_ Chapter II with equivalence of classes (where they are sets)   frege53c 37729
*20.26.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 37747
*20.26.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 37754
*20.26.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 37777
*20.26.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 37793
*20.26.4  Exploring Topology via Seifert And Threlfall   enrelmap 37812
*20.26.4.1  Equinumerosity of sets of relations and maps   enrelmap 37812
*20.26.4.2  Generic Pseudoclosure Spaces, Pseudointeror Spaces, and Pseudoneighborhoods   sscon34b 37838
*20.26.4.3  Generic Neighborhood Spaces   gneispa 37949
*20.26.5  Exploring Higher Homotopy via Kerodon   k0004lem1 37966
*20.26.5.1  Simplicial Sets   k0004lem1 37966
20.27  Mathbox for Stanislas Polu
20.27.1  IMO Problems   wwlemuld 37975
20.27.1.1  IMO 1972 B2   wwlemuld 37975
*20.27.2  INT Inequalities Proof Generator   int-addcomd 37997
20.27.4  AM-GM (for k = 2,3,4)   gsumws3 38020
20.28  Mathbox for Steve Rodriguez
20.28.1  Miscellanea   nanorxor 38025
20.28.2  Ratio test for infinite series convergence and divergence   dvgrat 38032
20.28.3  Multiples   reldvds 38035
20.28.4  Function operations   caofcan 38043
20.28.5  Calculus   lhe4.4ex1a 38049
20.28.6  The generalized binomial coefficient operation   cbcc 38056
20.28.7  Binomial series   uzmptshftfval 38066
20.29  Mathbox for Andrew Salmon
20.29.1  Principia Mathematica * 10   pm10.12 38078
20.29.2  Principia Mathematica * 11   2alanimi 38092
20.29.3  Predicate Calculus   sbeqal1 38119
20.29.4  Principia Mathematica * 13 and * 14   pm13.13a 38129
20.29.5  Set Theory   elnev 38160
20.29.7  Geometry   cplusr 38182
*20.30  Mathbox for Alan Sare
20.30.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 38204
20.30.2  Supplementary unification deductions   bi1imp 38208
20.30.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 38228
20.30.4  What is Virtual Deduction?   wvd1 38306
20.30.5  Virtual Deduction Theorems   df-vd1 38307
20.30.6  Theorems proved using Virtual Deduction   trsspwALT 38567
20.30.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 38603
20.30.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 38671
20.30.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 38675
20.30.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 38682
*20.30.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 38685
20.31  Mathbox for Glauco Siliprandi
20.31.1  Miscellanea   evth2f 38696
20.31.2  Functions   unima 38855
20.31.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 38984
20.31.4  Real intervals   gtnelioc 39158
20.31.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 39248
20.31.7  Limits   clim1fr1 39269
20.31.8  Trigonometry   coseq0 39410
20.31.9  Continuous Functions   mulcncff 39416
20.31.10  Derivatives   dvsinexp 39460
20.31.11  Integrals   itgsin0pilem1 39502
20.31.12  Stone Weierstrass theorem - real version   stoweidlem1 39555
20.31.13  Wallis' product for π   wallispilem1 39619
20.31.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 39628
20.31.15  Dirichlet kernel   dirkerval 39645
20.31.16  Fourier Series   fourierdlem1 39662
20.31.17  e is transcendental   elaa2lem 39787
20.31.18  n-dimensional Euclidean space   rrxtopn 39838
20.31.19  Basic measure theory   csalg 39865
*20.31.19.1  σ-Algebras   csalg 39865
20.31.19.2  Sum of nonnegative extended reals   csumge0 39916
*20.31.19.3  Measures   cmea 40003
*20.31.19.4  Outer measures and Caratheodory's construction   come 40040
*20.31.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 40087
*20.31.19.6  Measurable functions   csmblfn 40246
20.32  Mathbox for Saveliy Skresanov
20.32.1  Ceva's theorem   sigarval 40373
20.33  Mathbox for Jarvin Udandy
20.34  Mathbox for Alexander van der Vekens
20.34.1  Double restricted existential uniqueness   r19.32 40501
20.34.1.1  Restricted quantification (extension)   r19.32 40501
20.34.1.2  The empty set (extension)   raaan2 40509
20.34.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 40510
20.34.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 40515
*20.34.2  Alternative definitions of function and operation values   wdfat 40527
20.34.2.1  Restricted quantification (extension)   ralbinrald 40533
20.34.2.2  The universal class (extension)   nvelim 40534
20.34.2.3  Introduce the Axiom of Power Sets (extension)   alneu 40535
20.34.2.4  Relations (extension)   eldmressn 40537
20.34.2.5  Functions (extension)   fveqvfvv 40538
20.34.2.6  Predicate "defined at"   dfateq12d 40543
20.34.2.7  Alternative definition of the value of a function   dfafv2 40546
20.34.2.8  Alternative definition of the value of an operation   aoveq123d 40592
20.34.3  General auxiliary theorems   ralralimp 40622
20.34.3.1  The empty set - extension   ralralimp 40622
20.34.3.2  Unordered and ordered pairs - extension   elprneb 40623
20.34.3.3  Indexed union and intersection - extension   otiunsndisjX 40625
20.34.3.4  Functions - extension   fvifeq 40626
20.34.3.5  Ordering on reals - extension   leltletr 40635
20.34.3.6  Subtraction - extension   cnambpcma 40636
20.34.3.7  Ordering on reals (cont.) - extension   leaddsuble 40638
20.34.3.8  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 40644
20.34.3.9  Integers (as a subset of complex numbers) - extension   zgeltp1eq 40645
20.34.3.10  Decimal arithmetic - extension   1t10e1p1e11 40646
20.34.3.11  Upper sets of integers - extension   eluzge0nn0 40649
20.34.3.12  Infinity and the extended real number system (cont.) - extension   nltle2tri 40650
20.34.3.13  Finite intervals of integers - extension   ssfz12 40651
20.34.3.14  Half-open integer ranges - extension   fzopred 40659
20.34.3.15  The modulo (remainder) operation - extension   m1mod0mod1 40667
20.34.3.16  The infinite sequence builder "seq"   smonoord 40669
20.34.3.17  Finite and infinite sums - extension   fsummsndifre 40670
20.34.3.18  Extensible structures - extension   setsidel 40674
*20.34.4  Partitions of real intervals   ciccp 40677
20.34.5  Shifting functions with an integer range domain   fargshiftfv 40703
20.34.6  Words over a set (extension)   lswn0 40708
20.34.6.1  Last symbol of a word - extension   lswn0 40708
20.34.6.2  Concatenations with singleton words - extension   ccatw2s1cl 40709
*20.34.6.3  Prefixes of a word   cpfx 40710
20.34.7  Number theory (extension)   cfmtno 40768
*20.34.7.1  Fermat numbers   cfmtno 40768
*20.34.7.2  Mersenne primes   m2prm 40834
20.34.7.3  Proth's theorem   modexp2m1d 40858
*20.34.8  Even and odd numbers   ceven 40866
20.34.8.1  Definitions and basic properties   ceven 40866
20.34.8.2  Alternate definitions using the "divides" relation   dfeven2 40891
20.34.8.3  Alternate definitions using the "modulo" operation   dfeven3 40899
20.34.8.4  Alternate definitions using the "gcd" operation   iseven5 40905
20.34.8.5  Theorems of part 5 revised   zneoALTV 40909
20.34.8.6  Theorems of part 6 revised   odd2np1ALTV 40914
20.34.8.7  Theorems of AV's mathbox revised   0evenALTV 40928
20.34.8.9  Perfect Number Theorem (revised)   perfectALTVlem1 40955
*20.34.8.10  Goldbach's conjectures   cgbe 40958
20.34.9  Graph theory (extension)   1hegrlfgr 41031
20.34.9.1  Loop-free graphs - extension   1hegrlfgr 41031
20.34.9.2  Walks - extension   cupwlks 41032
20.34.10  Set of unordered pairs   sprid 41042
20.34.11  Monoids (extension)   ovn0dmfun 41082
20.34.11.1  Auxiliary theorems   ovn0dmfun 41082
20.34.11.2  Magmas and Semigroups (extension)   plusfreseq 41090
20.34.11.3  Magma homomorphisms and submagmas   cmgmhm 41095
20.34.11.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpt2ismgm 41125
*20.34.12  Magmas and internal binary operations (alternate approach)   ccllaw 41137
*20.34.12.1  Laws for internal binary operations   ccllaw 41137
*20.34.12.2  Internal binary operations   cintop 41150
20.34.12.3  Alternative definitions for Magmas and Semigroups   cmgm2 41169
20.34.13  Categories (extension)   idfusubc0 41183
20.34.13.1  Subcategories (extension)   idfusubc0 41183
20.34.14  Rings (extension)   lmod0rng 41186
20.34.14.1  Nonzero rings (extension)   lmod0rng 41186
*20.34.14.2  Non-unital rings ("rngs")   crng 41192
20.34.14.3  Rng homomorphisms   crngh 41203
20.34.14.4  Ring homomorphisms (extension)   rhmfn 41236
20.34.14.5  Ideals as non-unital rings   lidldomn1 41239
20.34.14.6  The non-unital ring of even integers   0even 41249
20.34.14.7  A constructed not unital ring   cznrnglem 41271
*20.34.14.8  The category of non-unital rings   crngc 41275
*20.34.14.9  The category of (unital) rings   cringc 41321
20.34.14.10  Subcategories of the category of rings   srhmsubclem1 41391
20.34.15  Basic algebraic structures (extension)   xpprsng 41428
20.34.15.1  Auxiliary theorems   xpprsng 41428
20.34.15.2  The binomial coefficient operation (extension)   bcpascm1 41447
20.34.15.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 41450
20.34.15.4  Ordered group sum operation (extension)   gsumpr 41457
20.34.15.5  Symmetric groups (extension)   exple2lt6 41463
20.34.15.6  Divisibility (extension)   invginvrid 41466
20.34.15.7  The support of functions (extension)   rmsupp0 41467
20.34.15.8  Finitely supported functions (extension)   rmsuppfi 41472
20.34.15.9  Left modules (extension)   lmodvsmdi 41481
20.34.15.10  Associative algebras (extension)   ascl0 41483
20.34.15.11  Univariate polynomials (extension)   ply1vr1smo 41487
20.34.15.12  Univariate polynomials (examples)   linply1 41499
20.34.16  Linear algebra (extension)   cdmatalt 41503
*20.34.16.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 41503
*20.34.16.2  Linear combinations   clinc 41511
*20.34.16.3  Linear independency   clininds 41547
20.34.16.4  Simple left modules and the ` ZZ `-module   lmod1lem1 41594
20.34.16.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 41614
20.34.17  Complexity theory   offval0 41617
20.34.17.1  Auxiliary theorems   offval0 41617
20.34.17.2  The modulo (remainder) operation (extension)   fldivmod 41631
20.34.17.3  Even and odd integers   nn0onn0ex 41636
20.34.17.4  The natural logarithm on complex numbers (extension)   logge0b 41647
20.34.17.5  Division of functions   cfdiv 41653
20.34.17.6  Upper bounds   cbigo 41663
20.34.17.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 41674
*20.34.17.8  The binary logarithm   fldivexpfllog2 41681
20.34.17.9  Binary length   cblen 41685
*20.34.17.10  Digits   cdig 41711
20.34.17.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 41731
20.34.17.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 41740
20.35  Mathbox for Emmett Weisz
*20.35.1  Miscellaneous Theorems   nfintd 41742
20.35.2  Set Recursion   csetrecs 41753
*20.35.2.1  Basic Properties of Set Recursion   csetrecs 41753
20.35.2.2  Examples and properties of set recursion   elsetrecslem 41767
*20.35.3  Construction of Games and Surreal Numbers   cpg 41775
*20.36  Mathbox for David A. Wheeler
*20.36.2  Greater than, greater than or equal to.   cge-real 41784
*20.36.3  Hyperbolic trigonometric functions   csinh 41794
*20.36.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 41805
*20.36.5  Identities for "if"   ifnmfalse 41827
*20.36.6  Decimal point   cdp2 41828
*20.36.7  Logarithms generalized to arbitrary base using ` logb `   logb2aval 41838
*20.36.8  Logarithm laws generalized to an arbitrary base - log_   clog- 41839
*20.36.9  Formally define terms such as Reflexivity   wreflexive 41841