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  Complex metric 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  Undirected graphs - basics
16.2  Eulerian paths and the Konigsberg Bridge problem
16.3  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 David A. Wheeler
20.36  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 110
*1.2.5  Logical equivalence   wb 194
*1.2.6  Logical disjunction and conjunction   wo 381
*1.2.7  Miscellaneous theorems of propositional calculus   pm5.62 959
*1.2.8  The conditional operator for propositions   wif 1005
*1.2.9  The weak deduction theorem   elimh 1023
1.2.10  Abbreviated conjunction and disjunction of three wff's   w3o 1029
1.2.11  Logical 'nand' (Sheffer stroke)   wnan 1438
1.2.12  Logical 'xor'   wxo 1455
1.2.13  True and false constants   wal 1472
*1.2.13.1  Universal quantifier for use by df-tru   wal 1472
*1.2.13.2  Equality predicate for use by df-tru   cv 1473
1.2.13.3  Define the true and false constants   wtru 1475
*1.2.14  Truth tables   truantru 1496
1.3  Other axiomatizations related to classical propositional calculus
*1.3.1  Minimal implicational calculus   minimp 1550
1.3.2  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1556
1.3.3  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1573
*1.3.4  Derive Nicod's axiom from the standard axioms   nic-dfim 1584
1.3.5  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1590
1.3.6  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1609
1.3.7  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1613
1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1628
1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1651
1.3.10  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1664
*1.3.11  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1683
*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 1694
1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1700
1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1713
1.4.4  Axiom scheme ax-5 (Distinctness) - first use of \$d   ax-5 1793
*1.4.5  Equality predicate (continued)   weq 1824
1.4.6  Define proper substitution   wsb 1830
1.4.7  Axiom scheme ax-6 (Existence)   ax-6 1838
1.4.8  Axiom scheme ax-7 (Equality)   ax-7 1885
1.4.9  Membership predicate   wcel 1938
1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 1940
1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 1947
*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 1953
*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 1966
1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 1971
1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 1983
1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2137
1.6  Existential uniqueness
1.7  Other axiomatizations related to classical predicate calculus
*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2455
*1.7.2  Intuitionistic logic   axia1 2479
*PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
2.1.1  Introduce the Axiom of Extensionality   ax-ext 2494
2.1.2  Class abstractions (a.k.a. class builders)   cab 2500
2.1.3  Class form not-free predicate   wnfc 2642
2.1.4  Negated equality and membership   wne 2684
2.1.4.1  Negated equality   neii 2688
2.1.4.2  Negated membership   neli 2789
2.1.5  Restricted quantification   wral 2800
2.1.6  The universal class   cvv 3077
*2.1.7  Conditional equality (experimental)   wcdeq 3289
2.1.9  Proper substitution of classes for sets   wsbc 3306
2.1.10  Proper substitution of classes for sets into classes   csb 3403
2.1.11  Define basic set operations and relations   cdif 3441
2.1.12  Subclasses and subsets   df-ss 3458
2.1.13  The difference, union, and intersection of two classes   difeq1 3587
2.1.13.1  The difference of two classes   difeq1 3587
2.1.13.2  The union of two classes   elun 3619
2.1.13.3  The intersection of two classes   elin 3662
2.1.13.4  The symmetric difference of two classes   csymdif 3708
2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 3719
2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 3756
2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 3769
2.1.14  The empty set   c0 3777
*2.1.15  "Weak deduction theorem" for set theory   cif 3939
2.1.16  Power classes   cpw 4011
2.1.17  Unordered and ordered pairs   snjust 4027
2.1.18  The union of a class   cuni 4270
2.1.19  The intersection of a class   cint 4308
2.1.20  Indexed union and intersection   ciun 4353
2.1.21  Disjointness   wdisj 4451
2.1.22  Binary relations   wbr 4481
2.1.23  Ordered-pair class abstractions (class builders)   copab 4540
2.1.24  Transitive classes   wtr 4578
2.2  ZF Set Theory - add the Axiom of Replacement
2.2.1  Introduce the Axiom of Replacement   ax-rep 4597
2.2.2  Derive the Axiom of Separation   axsep 4606
2.2.3  Derive the Null Set Axiom   zfnuleu 4612
2.2.4  Theorems requiring subset and intersection existence   nalset 4622
2.2.5  Theorems requiring empty set existence   class2set 4657
2.3  ZF Set Theory - add the Axiom of Power Sets
2.3.1  Introduce the Axiom of Power Sets   ax-pow 4668
2.3.2  Derive the Axiom of Pairing   zfpair 4730
2.3.3  Ordered pair theorem   opnz 4766
2.3.4  Ordered-pair class abstractions (cont.)   opabid 4801
2.3.5  Power class of union and intersection   pwin 4836
2.3.6  Epsilon and identity relations   cep 4841
2.3.7  Partial and complete ordering   wpo 4851
2.3.8  Founded and well-ordering relations   wfr 4888
2.3.9  Relations   cxp 4930
2.3.10  The Predecessor Class   cpred 5486
2.3.11  Well-founded induction   tz6.26 5518
2.3.12  Ordinals   word 5529
2.3.13  Definite description binder (inverted iota)   cio 5651
2.3.14  Functions   wfun 5683
2.3.15  Cantor's Theorem   canth 6384
2.3.16  Restricted iota (description binder)   crio 6386
2.3.17  Operations   co 6425
2.3.18  "Maps to" notation   mpt2ndm0 6648
2.3.19  Function operation   cof 6668
2.3.20  Proper subset relation   crpss 6709
2.4  ZF Set Theory - add the Axiom of Union
2.4.1  Introduce the Axiom of Union   ax-un 6722
2.4.2  Ordinals (continued)   ordon 6749
2.4.3  Transfinite induction   tfi 6820
2.4.4  The natural numbers (i.e. finite ordinals)   com 6832
2.4.5  Peano's postulates   peano1 6852
2.4.6  Finite induction (for finite ordinals)   find 6858
2.4.7  First and second members of an ordered pair   c1st 6931
*2.4.8  The support of functions   csupp 7056
*2.4.9  Special "Maps to" operations   opeliunxp2f 7097
2.4.10  Function transposition   ctpos 7112
2.4.11  Curry and uncurry   ccur 7152
2.4.12  Undefined values   cund 7159
2.4.13  Well-founded recursion   cwrecs 7167
2.4.14  Functions on ordinals; strictly monotone ordinal functions   iunon 7197
2.4.15  "Strong" transfinite recursion   crecs 7229
2.4.16  Recursive definition generator   crdg 7267
2.4.17  Finite recursion   frfnom 7292
2.4.18  Ordinal arithmetic   c1o 7315
2.4.19  Natural number arithmetic   nna0 7446
2.4.20  Equivalence relations and classes   wer 7501
2.4.21  The mapping operation   cmap 7619
2.4.22  Infinite Cartesian products   cixp 7669
2.4.23  Equinumerosity   cen 7713
2.4.24  Schroeder-Bernstein Theorem   sbthlem1 7830
2.4.25  Equinumerosity (cont.)   xpf1o 7882
2.4.26  Pigeonhole Principle   phplem1 7899
2.4.27  Finite sets   onomeneq 7910
2.4.28  Finitely supported functions   cfsupp 8033
2.4.29  Finite intersections   cfi 8074
2.4.30  Hall's marriage theorem   marypha1lem 8097
2.4.31  Supremum and infimum   csup 8104
2.4.32  Ordinal isomorphism, Hartog's theorem   coi 8172
2.4.33  Hartogs function, order types, weak dominance   char 8219
2.5  ZF Set Theory - add the Axiom of Regularity
2.5.1  Introduce the Axiom of Regularity   ax-reg 8255
2.5.2  Axiom of Infinity equivalents   inf0 8276
2.6  ZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity   ax-inf 8293
2.6.2  Existence of omega (the set of natural numbers)   omex 8298
2.6.3  Cantor normal form   ccnf 8316
2.6.4  Transitive closure   trcl 8362
2.6.5  Rank   cr1 8383
2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 8506
2.6.7  Cardinal numbers   ccrd 8519
2.6.8  Axiom of Choice equivalents   wac 8696
2.6.9  Cardinal number arithmetic   ccda 8747
2.6.10  The Ackermann bijection   ackbij2lem1 8799
2.6.11  Cofinality (without Axiom of Choice)   cflem 8826
2.6.12  Eight inequivalent definitions of finite set   sornom 8857
2.6.13  Hereditarily size-limited sets without Choice   itunifval 8996
*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 9015
3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9026
3.2  ZFC Set Theory - add the Axiom of Choice
3.2.1  Introduce the Axiom of Choice   ax-ac 9039
3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9074
3.2.3  Cardinal number theorems using Axiom of Choice   cardval 9122
3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 9150
3.2.5  Cofinality using Axiom of Choice   alephreg 9158
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 9196
3.4.2  Derivation of the Axiom of Choice   gchaclem 9254
*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
4.1  Inaccessibles
4.1.1  Weakly and strongly inaccessible cardinals   cwina 9258
4.1.2  Weak universes   cwun 9276
4.1.3  Tarski classes   ctsk 9324
4.1.4  Grothendieck universes   cgru 9366
4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 9399
4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 9402
4.2.3  Tarski map function   ctskm 9413
*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 9420
5.1.2  Final derivation of real and complex number postulates   axaddf 9720
5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9746
5.2  Derive the basic properties from the field axioms
5.2.1  Some deductions from the field axioms for complex numbers   cnex 9771
5.2.2  Infinity and the extended real number system   cpnf 9825
5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9858
5.2.4  Ordering on reals   lttr 9863
5.2.5  Initial properties of the complex numbers   mul12 9952
5.3  Real and complex numbers - basic operations
5.3.2  Subtraction   cmin 10016
5.3.3  Multiplication   kcnktkm1cn 10211
5.3.4  Ordering on reals (cont.)   gt0ne0 10241
5.3.5  Reciprocals   ixi 10404
5.3.6  Division   cdiv 10432
5.3.7  Ordering on reals (cont.)   elimgt0 10607
5.3.8  Completeness Axiom and Suprema   fimaxre 10717
5.3.9  Imaginary and complex number properties   inelr 10764
5.3.10  Function operation analogue theorems   ofsubeq0 10771
5.4  Integer sets
5.4.1  Positive integers (as a subset of complex numbers)   cn 10774
5.4.2  Principle of mathematical induction   nnind 10792
*5.4.3  Decimal representation of numbers   c2 10824
*5.4.4  Some properties of specific numbers   neg1cn 10878
5.4.5  Simple number properties   halfcl 11011
5.4.6  The Archimedean property   nnunb 11042
5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11046
5.4.8  Integers (as a subset of complex numbers)   cz 11117
5.4.9  Decimal arithmetic   cdc 11232
5.4.10  Upper sets of integers   cuz 11426
5.4.11  Well-ordering principle for bounded-below sets of integers   uzwo3 11524
5.4.12  Rational numbers (as a subset of complex numbers)   cq 11529
5.4.13  Existence of the set of complex numbers   rpnnen1lem2 11555
5.5  Order sets
5.5.1  Positive reals (as a subset of complex numbers)   crp 11573
5.5.2  Infinity and the extended real number system (cont.)   cxne 11684
5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 11872
5.5.4  Real number intervals   cioo 11914
5.5.5  Finite intervals of integers   cfz 12064
*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 12167
5.5.7  Half-open integer ranges   cfzo 12201
5.6  Elementary integer functions
5.6.1  The floor and ceiling functions   cfl 12320
5.6.2  The modulo (remainder) operation   cmo 12397
5.6.3  Miscellaneous theorems about integers   om2uz0i 12475
5.6.4  Strong induction over upper sets of integers   uzsinds 12515
5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 12518
5.6.6  The infinite sequence builder "seq"   cseq 12530
5.6.7  Integer powers   cexp 12589
5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 12783
5.6.9  Factorial function   cfa 12789
5.6.10  The binomial coefficient operation   cbc 12818
5.6.11  The ` # ` (set size) function   chash 12846
5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 12970
5.6.11.2  Finite induction on the size of the first component of a binary relation   brfi1indlem 12990
*5.7  Words over a set
5.7.1  Definitions and basic theorems   cword 13003
5.7.2  Last symbol of a word   lsw 13061
5.7.3  Concatenations of words   ccatfn 13067
5.7.4  Singleton words   ids1 13087
5.7.5  Concatenations with singleton words   ccatws1cl 13106
5.7.6  Subwords   swrdval 13126
5.7.7  Subwords of subwords   swrdswrdlem 13168
5.7.8  Subwords and concatenations   wrdcctswrd 13174
5.7.9  Subwords of concatenations   swrdccatfn 13190
5.7.10  Splicing words (substring replacement)   splval 13210
5.7.11  Reversing words   revval 13217
5.7.12  Repeated symbol words   reps 13225
*5.7.13  Cyclical shifts of words   ccsh 13242
5.7.14  Mapping words by a function   wrdco 13285
5.7.15  Longer string literals   cs2 13294
*5.8  Reflexive and transitive closures of relations
5.8.1  The reflexive and transitive properties of relations   coss12d 13416
5.8.2  Basic properties of closures   cleq1lem 13426
5.8.3  Definitions and basic properties of transitive closures   ctcl 13429
5.8.4  Exponentiation of relations   crelexp 13465
5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 13500
*5.8.6  Principle of transitive induction.   relexpindlem 13508
5.9  Elementary real and complex functions
5.9.1  The "shift" operation   cshi 13511
5.9.2  Signum (sgn or sign) function   csgn 13531
5.9.3  Real and imaginary parts; conjugate   ccj 13541
5.9.4  Square root; absolute value   csqrt 13678
5.10  Elementary limits and convergence
5.10.1  Superior limit (lim sup)   clsp 13906
5.10.2  Limits   cli 13927
5.10.3  Finite and infinite sums   csu 14131
5.10.4  The binomial theorem   binomlem 14267
5.10.5  The inclusion/exclusion principle   incexclem 14274
5.10.6  Infinite sums (cont.)   isumshft 14277
5.10.7  Miscellaneous converging and diverging sequences   divrcnv 14290
5.10.8  Arithmetic series   arisum 14298
5.10.9  Geometric series   expcnv 14302
5.10.10  Ratio test for infinite series convergence   cvgrat 14321
5.10.11  Mertens' theorem   mertenslem1 14322
5.10.12  Finite and infinite products   prodf 14325
5.10.12.1  Product sequences   prodf 14325
5.10.12.2  Non-trivial convergence   ntrivcvg 14335
5.10.12.3  Complex products   cprod 14341
5.10.12.4  Finite products   fprod 14377
5.10.12.5  Infinite products   iprodclim 14435
5.10.13  Falling and Rising Factorial   cfallfac 14441
5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 14483
5.11  Elementary trigonometry
5.11.1  The exponential, sine, and cosine functions   ce 14498
5.11.2  _e is irrational   eirrlem 14638
5.12  Cardinality of real and complex number subsets
5.12.1  Countability of integers and rationals   xpnnen 14645
5.12.2  The reals are uncountable   rpnnen2lem1 14649
*PART 6  ELEMENTARY NUMBER THEORY
6.1  Elementary properties of divisibility
6.1.1  Irrationality of square root of 2   sqr2irrlem 14683
6.1.2  Some Number sets are chains of proper subsets   nthruc 14686
6.1.3  The divides relation   cdvds 14688
*6.1.4  Even and odd numbers   evenelz 14765
6.1.5  The division algorithm   divalglem0 14821
6.1.6  Bit sequences   cbits 14849
6.1.7  The greatest common divisor operator   cgcd 14925
6.1.8  Bézout's identity   bezoutlem1 14965
6.1.9  Algorithms   nn0seqcvgd 14995
6.1.10  Euclid's Algorithm   eucalgval2 15006
*6.1.11  The least common multiple   clcm 15013
*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 15074
6.1.13  Cancellability of congruences   congr 15090
6.2  Elementary prime number theory
*6.2.1  Elementary properties   cprime 15097
*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 15135
6.2.3  Properties of the canonical representation of a rational   cnumer 15153
6.2.4  Euler's theorem   codz 15180
6.2.5  Arithmetic modulo a prime number   modprm1div 15222
6.2.6  Pythagorean Triples   coprimeprodsq 15233
6.2.7  The prime count function   cpc 15261
6.2.8  Pocklington's theorem   prmpwdvds 15328
6.2.9  Infinite primes theorem   unbenlem 15332
6.2.10  Sum of prime reciprocals   prmreclem1 15340
6.2.11  Fundamental theorem of arithmetic   1arithlem1 15347
6.2.12  Lagrange's four-square theorem   cgz 15353
6.2.13  Van der Waerden's theorem   cvdwa 15389
6.2.14  Ramsey's theorem   cram 15423
*6.2.15  Primorial function   cprmo 15455
*6.2.16  Prime gaps   prmgaplem1 15473
6.2.17  Decimal arithmetic (cont.)   dec2dvds 15487
6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 15520
6.2.19  Specific prime numbers   prmlem0 15532
6.2.20  Very large primes   1259lem1 15558
PART 7  BASIC STRUCTURES
7.1  Extensible structures
*7.1.1  Basic definitions   cstr 15573
7.1.2  Slot definitions   cplusg 15650
7.1.3  Definition of the structure product   crest 15786
7.1.4  Definition of the structure quotient   cordt 15864
7.2  Moore spaces
7.2.1  Moore closures   mrcflem 15979
7.2.2  Independent sets in a Moore system   mrisval 16003
7.2.3  Algebraic closure systems   isacs 16025
PART 8  BASIC CATEGORY THEORY
8.1  Categories
8.1.1  Categories   ccat 16038
8.1.2  Opposite category   coppc 16084
8.1.3  Monomorphisms and epimorphisms   cmon 16101
8.1.4  Sections, inverses, isomorphisms   csect 16117
*8.1.5  Isomorphic objects   ccic 16168
8.1.6  Subcategories   cssc 16180
8.1.7  Functors   cfunc 16227
8.1.8  Full & faithful functors   cful 16275
8.1.9  Natural transformations and the functor category   cnat 16314
8.1.10  Initial, terminal and zero objects of a category   cinito 16351
8.2  Arrows (disjointified hom-sets)
8.2.1  Identity and composition for arrows   cida 16416
8.3  Examples of categories
8.3.1  The category of sets   csetc 16438
8.3.2  The category of categories   ccatc 16457
*8.3.3  The category of extensible structures   fncnvimaeqv 16473
8.4  Categorical constructions
8.4.1  Product of categories   cxpc 16521
8.4.2  Functor evaluation   cevlf 16562
8.4.3  Hom functor   chof 16601
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 16653
9.2.2  Lattices   clat 16758
9.2.3  The dual of an ordered set   codu 16841
9.2.4  Subset order structures   cipo 16864
9.2.5  Distributive lattices   latmass 16901
9.2.6  Posets and lattices as relations   cps 16911
9.2.7  Directed sets, nets   cdir 16941
PART 10  BASIC ALGEBRAIC STRUCTURES
10.1  Monoids
*10.1.1  Magmas   cplusf 16952
*10.1.2  Identity elements   mgmidmo 16972
*10.1.3  Ordered sums in a magma   gsumvalx 16983
*10.1.4  Semigroups   csgrp 16996
*10.1.5  Definition and basic properties of monoids   cmnd 17007
10.1.6  Monoid homomorphisms and submonoids   cmhm 17046
*10.1.7  Ordered sums in a monoid   gsumvallem2 17085
10.1.8  Free monoids   cfrmd 17097
10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 17118
10.2  Groups
10.2.1  Definition and basic properties   cgrp 17135
*10.2.2  Group multiple operation   cmg 17253
10.2.3  Subgroups and Quotient groups   csubg 17301
10.2.4  Elementary theory of group homomorphisms   cghm 17370
10.2.5  Isomorphisms of groups   cgim 17412
10.2.6  Group actions   cga 17435
10.2.7  Centralizers and centers   ccntz 17461
10.2.8  The opposite group   coppg 17488
10.2.9  Symmetric groups   csymg 17510
*10.2.9.1  Definition and basic properties   csymg 17510
10.2.9.2  Cayley's theorem   cayleylem1 17545
10.2.9.3  Permutations fixing one element   symgfix2 17549
*10.2.9.4  Transpositions in the symmetric group   cpmtr 17574
10.2.9.5  The sign of a permutation   cpsgn 17622
10.2.10  p-Groups and Sylow groups; Sylow's theorems   cod 17657
10.2.11  Direct products   clsm 17778
10.2.12  Free groups   cefg 17848
10.3  Abelian groups
10.3.1  Definition and basic properties   ccmn 17922
10.3.2  Cyclic groups   ccyg 18007
10.3.3  Group sum operation   gsumval3a 18032
10.3.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 18107
10.3.5  Internal direct products   cdprd 18120
10.3.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 18192
10.4  Rings
10.4.1  Multiplicative Group   cmgp 18217
10.4.2  Ring unit   cur 18229
10.4.2.1  Semirings   csrg 18233
*10.4.2.2  The binomial theorem for semirings   srgbinomlem1 18268
10.4.3  Definition and basic properties of unital rings   crg 18275
10.4.4  Opposite ring   coppr 18350
10.4.5  Divisibility   cdsr 18366
10.4.6  Ring homomorphisms   crh 18440
10.5  Division rings and fields
10.5.1  Definition and basic properties   cdr 18475
10.5.2  Subrings of a ring   csubrg 18504
10.5.3  Absolute value (abstract algebra)   cabv 18544
10.5.4  Star rings   cstf 18571
10.6  Left modules
10.6.1  Definition and basic properties   clmod 18591
10.6.2  Subspaces and spans in a left module   clss 18655
10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 18742
10.6.4  Subspace sum; bases for a left module   clbs 18797
10.7  Vector spaces
10.7.1  Definition and basic properties   clvec 18825
10.8  Ideals
10.8.1  The subring algebra; ideals   csra 18891
10.8.2  Two-sided ideals and quotient rings   c2idl 18954
10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 18964
10.8.4  Nonzero rings and zero rings   cnzr 18980
10.8.5  Left regular elements. More kinds of rings   crlreg 19002
10.9  Associative algebras
10.9.1  Definition and basic properties   casa 19032
10.10  Abstract multivariate polynomials
10.10.1  Definition and basic properties   cmps 19074
10.10.2  Polynomial evaluation   ces 19227
*10.10.3  Additional definitions for (multivariate) polynomials   cmhp 19260
*10.10.4  Univariate polynomials   cps1 19268
10.10.5  Univariate polynomial evaluation   ces1 19401
10.11  The complex numbers as an algebraic extensible structure
10.11.1  Definition and basic properties   cpsmet 19453
*10.11.2  Ring of integers   zring 19539
10.11.3  Algebraic constructions based on the complex numbers   czrh 19571
10.11.4  Signs as subgroup of the complex numbers   cnmsgnsubg 19646
10.11.5  Embedding of permutation signs into a ring   zrhpsgnmhm 19653
10.11.6  The ordered field of real numbers   crefld 19673
10.12  Generalized pre-Hilbert and Hilbert spaces
10.12.1  Definition and basic properties   cphl 19692
10.12.2  Orthocomplements and closed subspaces   cocv 19724
10.12.3  Orthogonal projection and orthonormal bases   cpj 19764
*PART 11  BASIC LINEAR ALGEBRA
11.1  Vectors and free modules
*11.1.1  Direct sum of left modules   cdsmm 19795
*11.1.2  Free modules   cfrlm 19810
*11.1.3  Standard basis (unit vectors)   cuvc 19841
*11.1.4  Independent sets and families   clindf 19863
11.1.5  Characterization of free modules   lmimlbs 19895
*11.2  Matrices
*11.2.1  The matrix multiplication   cmmul 19909
*11.2.2  Square matrices   cmat 19933
*11.2.3  The matrix algebra   matmulr 19964
*11.2.4  Matrices of dimension 0 and 1   mat0dimbas0 19992
*11.2.5  The subalgebras of diagonal and scalar matrices   cdmat 20014
*11.2.6  Multiplication of a matrix with a "column vector"   cmvmul 20066
11.2.7  Replacement functions for a square matrix   cmarrep 20082
11.2.8  Submatrices   csubma 20102
11.3  The determinant
11.3.1  Definition and basic properties   cmdat 20110
11.3.2  Determinants of 2 x 2 -matrices   m2detleiblem1 20150
*11.3.4  Laplace expansion of determinants (special case)   symgmatr01lem 20179
11.3.5  Inverse matrix   invrvald 20202
*11.3.6  Cramer's rule   slesolvec 20205
*11.4  Polynomial matrices
11.4.1  Basic properties   pmatring 20218
*11.4.2  Constant polynomial matrices   ccpmat 20228
*11.4.3  Collecting coefficients of polynomial matrices   cdecpmat 20287
*11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 20317
*11.5  The characteristic polynomial
*11.5.1  Definition and basic properties   cchpmat 20351
*11.5.2  The characteristic factor function G   fvmptnn04if 20374
*11.5.3  The Cayley-Hamilton theorem   cpmadurid 20392
PART 12  BASIC TOPOLOGY
12.1  Topology
12.1.1  Topological spaces   ctop 20418
12.1.2  TopBases for topologies   isbasisg 20463
12.1.3  Examples of topologies   distop 20511
12.1.4  Closure and interior   ccld 20531
12.1.5  Neighborhoods   cnei 20612
12.1.6  Limit points and perfect sets   clp 20649
12.1.7  Subspace topologies   restrcl 20672
12.1.8  Order topology   ordtbaslem 20703
12.1.9  Limits and continuity in topological spaces   ccn 20739
12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 20821
12.1.11  Compactness   ccmp 20900
12.1.12  Bolzano-Weierstrass theorem   bwth 20924
12.1.13  Connectedness   ccon 20925
12.1.14  First- and second-countability   c1stc 20951
12.1.15  Local topological properties   clly 20978
12.1.16  Refinements   cref 21016
12.1.17  Compactly generated spaces   ckgen 21047
12.1.18  Product topologies   ctx 21074
12.1.19  Continuous function-builders   cnmptid 21175
12.1.20  Quotient maps and quotient topology   ckq 21207
12.1.21  Homeomorphisms   chmeo 21267
12.2  Filters and filter bases
12.2.1  Filter bases   elmptrab 21341
12.2.2  Filters   cfil 21360
12.2.3  Ultrafilters   cufil 21414
12.2.4  Filter limits   cfm 21448
12.2.5  Extension by continuity   ccnext 21574
12.2.6  Topological groups   ctmd 21585
12.2.7  Infinite group sum on topological groups   ctsu 21640
12.2.8  Topological rings, fields, vector spaces   ctrg 21670
12.3  Uniform Structures and Spaces
12.3.1  Uniform structures   cust 21714
12.3.2  The topology induced by an uniform structure   cutop 21745
12.3.3  Uniform Spaces   cuss 21768
12.3.4  Uniform continuity   cucn 21790
12.3.5  Cauchy filters in uniform spaces   ccfilu 21801
12.3.6  Complete uniform spaces   ccusp 21812
12.4  Metric spaces
12.4.1  Pseudometric spaces   ispsmet 21820
12.4.2  Basic metric space properties   cxme 21832
12.4.3  Metric space balls   blfvalps 21898
12.4.4  Open sets of a metric space   mopnval 21953
12.4.5  Continuity in metric spaces   metcnp3 22055
12.4.6  The uniform structure generated by a metric   metuval 22064
12.4.7  Examples of metric spaces   dscmet 22087
12.4.8  Normed algebraic structures   cnm 22091
12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 22206
12.4.10  Topology on the reals   qtopbaslem 22279
12.4.11  Topological definitions using the reals   cii 22407
12.4.12  Path homotopy   chtpy 22498
12.4.13  The fundamental group   cpco 22532
12.5  Complex metric vector spaces
12.5.1  Complex left modules   cclm 22594
12.5.2  Complex vector spaces   ccvs 22636
12.5.3  Complex pre-Hilbert space   ccph 22645
12.5.4  Convergence and completeness   ccfil 22723
12.5.5  Baire's Category Theorem   bcthlem1 22793
12.5.6  Banach spaces and complex Hilbert spaces   ccms 22801
12.5.6.1  The complete ordered field of the real numbers   retopn 22839
12.5.7  Euclidean spaces   crrx 22843
12.5.8  Minimizing Vector Theorem   minveclem1 22867
12.5.9  Projection Theorem   pjthlem1 22892
PART 13  BASIC REAL AND COMPLEX ANALYSIS
13.1  Continuity
13.1.1  Intermediate value theorem   pmltpclem1 22900
13.2  Integrals
13.2.1  Lebesgue measure   covol 22914
13.2.2  Lebesgue integration   cmbf 23065
13.2.2.1  Lesbesgue integral   cmbf 23065
13.2.2.2  Lesbesgue directed integral   cdit 23292
13.3  Derivatives
13.3.1  Real and complex differentiation   climc 23308
13.3.1.1  Derivatives of functions of one complex or real variable   climc 23308
13.3.1.2  Results on real differentiation   dvferm1lem 23427
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
14.1  Polynomials
14.1.1  Polynomial degrees   cmdg 23493
14.1.2  The division algorithm for univariate polynomials   cmn1 23565
14.1.3  Elementary properties of complex polynomials   cply 23629
14.1.4  The division algorithm for polynomials   cquot 23734
14.1.5  Algebraic numbers   caa 23758
14.1.6  Liouville's approximation theorem   aalioulem1 23779
14.2  Sequences and series
14.2.1  Taylor polynomials and Taylor's theorem   ctayl 23799
14.2.2  Uniform convergence   culm 23822
14.2.3  Power series   pserval 23856
14.3  Basic trigonometry
14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 23889
14.3.2  Properties of pi = 3.14159...   pilem1 23897
14.3.3  Mapping of the exponential function   efgh 23979
14.3.4  The natural logarithm on complex numbers   clog 23993
*14.3.5  Logarithms to an arbitrary base   clogb 24190
14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 24219
14.3.8  Inverse trigonometric functions   casin 24277
14.3.9  The Birthday Problem   log2ublem1 24361
14.3.10  Areas in R^2   carea 24370
14.3.11  More miscellaneous converging sequences   rlimcnp 24380
14.3.12  Inequality of arithmetic and geometric means   cvxcl 24399
14.3.13  Euler-Mascheroni constant   cem 24406
14.3.14  Zeta function   czeta 24427
14.3.15  Gamma function   clgam 24430
14.4  Basic number theory
14.4.1  Wilson's theorem   wilthlem1 24482
14.4.2  The Fundamental Theorem of Algebra   ftalem1 24487
14.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 24497
14.4.4  Number-theoretical functions   ccht 24507
14.4.5  Perfect Number Theorem   mersenne 24642
14.4.6  Characters of Z/nZ   cdchr 24647
14.4.7  Bertrand's postulate   bcctr 24690
*14.4.8  Quadratic residues and the Legendre symbol   clgs 24709
*14.4.9  Gauss' Lemma   gausslemma2dlem0a 24771
14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 24832
14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 24848
14.4.13  The Prime Number Theorem   mudivsum 24909
14.4.14  Ostrowski's theorem   abvcxp 24994
*PART 15  ELEMENTARY GEOMETRY
15.1  Definition and Tarski's Axioms of Geometry
15.2  Tarskian Geometry
15.2.1  Congruence   tgcgrcomimp 25062
15.2.2  Betweenness   tgbtwntriv2 25072
15.2.3  Dimension   tglowdim1 25085
15.2.4  Betweenness and Congruence   tgifscgr 25094
15.2.5  Congruence of a series of points   ccgrg 25096
15.2.6  Motions   cismt 25118
15.2.7  Colinearity   tglng 25132
15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 25158
15.2.9  Less-than relation in geometric congruences   cleg 25168
15.2.10  Rays   chlg 25186
15.2.11  Lines   btwnlng1 25205
15.2.12  Point inversions   cmir 25238
15.2.13  Right angles   crag 25279
15.2.14  Half-planes   islnopp 25322
15.2.15  Midpoints and Line Mirroring   cmid 25355
15.2.16  Congruence of angles   ccgra 25390
15.2.17  Angle Comparisons   cinag 25417
15.2.18  Congruence Theorems   tgsas1 25426
15.2.19  Equilateral triangles   ceqlg 25436
15.3  Properties of geometries
15.3.1  Isomorphisms between geometries   f1otrgds 25440
15.4  Geometry in Hilbert spaces
15.4.1  Geometry in the complex plane   cchhllem 25458
15.4.2  Geometry in Euclidean spaces   cee 25459
15.4.2.1  Definition of the Euclidean space   cee 25459
15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 25484
15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 25548
*PART 16  GRAPH THEORY
16.1  Undirected graphs - basics
16.1.1  Undirected hypergraphs   cuhg 25558
16.1.2  Undirected multigraphs   cumg 25580
16.1.3  Undirected simple graphs   cuslg 25597
16.1.3.1  Undirected simple graphs - basics   cuslg 25597
16.1.3.2  Undirected simple graphs - examples   usgraex0elv 25663
16.1.3.3  Finite undirected simple graphs   fiusgraedgfi 25675
16.1.4  Neighbors, complete graphs and universal vertices   cnbgra 25685
16.1.4.1  Neighbors   nbgraop 25691
16.1.4.2  Complete graphs   iscusgra 25724
16.1.4.3  Universal vertices   isuvtx 25755
16.1.5  Walks, paths and cycles   cwalk 25765
16.1.5.1  Walks and trails   relwlk 25785
16.1.5.2  Paths and simple paths   pths 25835
16.1.5.3  Circuits and cycles   crcts 25889
16.1.5.4  Connected graphs   cconngra 25936
16.1.5.5  Walks as words   cwwlk 25944
16.1.5.6  Closed walks   cclwlk 26014
16.1.5.7  Walks/paths of length 2 as ordered triples   c2wlkot 26120
16.1.6  Vertex degree   cvdg 26159
16.1.7  Regular graphs   crgra 26188
16.1.7.1  Definition and basic properties   crgra 26188
16.1.7.2  Walks in regular graphs   rusgranumwwlkl1 26212
16.2  Eulerian paths and the Konigsberg Bridge problem
16.2.1  Eulerian paths   ceup 26228
16.2.2  The Konigsberg Bridge problem   vdeg0i 26248
*16.3  The Friendship Theorem
16.3.1  Friendship graphs - basics   cfrgra 26254
16.3.2  The friendship theorem for small graphs   frgra1v 26264
16.3.3  Theorems according to Mertzios and Unger   2pthfrgrarn 26275
*16.3.4  Huneke's Proof of the Friendship Theorem   frgrancvvdeqlem1 26296
PART 17  GUIDES AND MISCELLANEA
17.1  Guides (conventions, explanations, and examples)
*17.1.1  Conventions   conventions 26389
17.1.2  Natural deduction   natded 26391
*17.1.3  Natural deduction examples   ex-natded5.2 26392
17.1.4  Definitional examples   ex-or 26409
17.1.5  Other examples   aevdemo 26448
17.2  Humor
17.2.1  April Fool's theorem   avril1 26450
17.3  (Future - to be reviewed and classified)
17.3.1  Planar incidence geometry   cplig 26457
17.3.2  Algebra preliminaries   crpm 26462
*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 26466
18.1.2  Abelian groups   cablo 26524
18.2  Complex vector spaces
18.2.1  Definition and basic properties   cvc 26539
18.2.2  Examples of complex vector spaces   cnaddablo 26577
18.3  Normed complex vector spaces
18.3.1  Definition and basic properties   cnv 26580
18.3.2  Examples of normed complex vector spaces   cnnv 26685
18.3.3  Induced metric of a normed complex vector space   imsval 26694
18.3.4  Inner product   cdip 26713
18.3.5  Subspaces   css 26737
18.4  Operators on complex vector spaces
18.4.1  Definitions and basic properties   clno 26758
18.5  Inner product (pre-Hilbert) spaces
18.5.1  Definition and basic properties   ccphlo 26830
18.5.2  Examples of pre-Hilbert spaces   cncph 26837
18.5.3  Properties of pre-Hilbert spaces   isph 26840
18.6  Complex Banach spaces
18.6.1  Definition and basic properties   ccbn 26881
18.6.2  Examples of complex Banach spaces   cnbn 26888
18.6.3  Uniform Boundedness Theorem   ubthlem1 26889
18.6.4  Minimizing Vector Theorem   minvecolem1 26893
18.7  Complex Hilbert spaces
18.7.1  Definition and basic properties   chlo 26914
18.7.2  Standard axioms for a complex Hilbert space   hlex 26927
18.7.3  Examples of complex Hilbert spaces   cnchl 26945
18.7.4  Subspaces   ssphl 26946
18.7.5  Hellinger-Toeplitz Theorem   htthlem 26947
*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
19.1  Axiomatization of complex pre-Hilbert spaces
19.1.1  Basic Hilbert space definitions   chil 26949
19.1.2  Preliminary ZFC lemmas   df-hnorm 26998
*19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 27011
*19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 27029
19.1.5  Vector operations   hvmulex 27041
19.1.6  Inner product postulates for a Hilbert space   ax-hfi 27109
19.2  Inner product and norms
19.2.1  Inner product   his5 27116
19.2.2  Norms   dfhnorm2 27152
19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 27190
19.3  Cauchy sequences and completeness axiom
19.3.1  Cauchy sequences and limits   hcau 27214
19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 27224
19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 27232
19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 27233
19.4  Subspaces and projections
19.4.1  Subspaces   df-sh 27237
19.4.2  Closed subspaces   df-ch 27251
19.4.3  Orthocomplements   df-oc 27282
19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 27340
19.4.5  Projection theorem   pjhthlem1 27423
19.4.6  Projectors   df-pjh 27427
19.5  Properties of Hilbert subspaces
19.5.1  Orthomodular law   omlsilem 27434
19.5.2  Projectors (cont.)   pjhtheu2 27448
19.5.3  Hilbert lattice operations   sh0le 27472
19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 27573
19.5.5  Commutes relation for Hilbert lattice elements   df-cm 27615
19.5.6  Foulis-Holland theorem   fh1 27650
19.5.7  Quantum Logic Explorer axioms   qlax1i 27659
19.5.8  Orthogonal subspaces   chscllem1 27669
19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 27686
19.5.10  Projectors (cont.)   pjorthi 27701
19.5.11  Mayet's equation E_3   mayete3i 27760
19.6  Operators on Hilbert spaces
*19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 27762
19.6.2  Zero and identity operators   df-h0op 27780
19.6.3  Operations on Hilbert space operators   hoaddcl 27790
19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 27871
19.6.5  Linear and continuous functionals and norms   df-nmfn 27877
19.6.7  Dirac bra-ket notation   df-bra 27882
19.6.8  Positive operators   df-leop 27884
19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 27885
19.6.10  Theorems about operators and functionals   nmopval 27888
19.6.11  Riesz lemma   riesz3i 28094
19.6.13  Quantum computation error bound theorem   unierri 28136
19.6.14  Dirac bra-ket notation (cont.)   branmfn 28137
19.6.15  Positive operators (cont.)   leopg 28154
19.6.16  Projectors as operators   pjhmopi 28178
19.7  States on a Hilbert lattice and Godowski's equation
19.7.1  States on a Hilbert lattice   df-st 28243
19.7.2  Godowski's equation   golem1 28303
19.8  Cover relation, atoms, exchange axiom, and modular symmetry
19.8.1  Covers relation; modular pairs   df-cv 28311
19.8.2  Atoms   df-at 28370
19.8.3  Superposition principle   superpos 28386
19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 28387
19.8.5  Irreducibility   chirredlem1 28422
19.8.6  Atoms (cont.)   atcvat3i 28428
19.8.7  Modular symmetry   mdsymlem1 28435
PART 20  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
20.1  Mathboxes for user contributions
20.1.1  Mathbox guidelines   mathbox 28474
20.2  Mathbox for Stefan Allan
20.3  Mathbox for Thierry Arnoux
20.3.1  Propositional Calculus - misc additions   bian1d 28479
20.3.2  Predicate Calculus   spc2ed 28485
20.3.2.1  Predicate Calculus - misc additions   spc2ed 28485
20.3.2.2  Restricted quantification - misc additions   ralcom4f 28489
20.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 28493
20.3.2.4  Existential "at most one" - misc additions   moel 28496
20.3.2.5  Existential uniqueness - misc additions   2reuswap2 28501
20.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 28505
20.3.3  General Set Theory   rabrab 28511
20.3.3.1  Class abstractions (a.k.a. class builders)   rabrab 28511
20.3.3.2  Image Sets   abrexdomjm 28518
20.3.3.3  Set relations and operations - misc additions   eqri 28524
20.3.3.4  Unordered pairs   elpreq 28533
20.3.3.5  Conditional operator - misc additions   ifeqeqx 28534
20.3.3.6  Set union   uniinn0 28538
20.3.3.7  Indexed union - misc additions   cbviunf 28544
20.3.3.8  Disjointness - misc additions   disjnf 28555
20.3.4  Relations and Functions   xpdisjres 28582
20.3.4.1  Relations - misc additions   xpdisjres 28582
20.3.4.2  Functions - misc additions   mptexgf 28598
20.3.4.3  Operations - misc additions   mpt2mptxf 28649
20.3.4.4  Isomorphisms - misc. add.   gtiso 28650
20.3.4.5  Disjointness (additional proof requiring functions)   disjdsct 28652
20.3.4.6  First and second members of an ordered pair - misc additions   df1stres 28653
20.3.4.7  Supremum - misc additions   supssd 28659
20.3.4.8  Finite Sets   imafi2 28661
20.3.4.9  Countable Sets   snct 28663
20.3.5  Real and Complex Numbers   addeq0 28687
20.3.5.3  Extended reals - misc additions   xgepnf 28693
20.3.5.4  Real number intervals - misc additions   joiniooico 28722
20.3.5.5  Finite intervals of integers - misc additions   nndiffz1 28732
20.3.5.6  Half-open integer ranges - misc additions   iundisjfi 28738
20.3.5.7  The ` # ` (set size) function - misc additions   hashunif 28745
20.3.5.8  The greatest common divisor operator - misc. add   numdenneg 28746
20.3.5.9  Integers   nnindf 28748
20.3.5.10  Division in the extended real number system   cxdiv 28752
20.3.6  Prime Number Theory   bhmafibid1 28771
20.3.6.1  Fermat's two square theorem   bhmafibid1 28771
20.3.7  Extensible Structures   ressplusf 28777
20.3.7.1  Structure restriction operator   ressplusf 28777
20.3.7.2  The opposite group   oppgle 28780
20.3.7.3  Posets   ressprs 28782
20.3.7.4  Complete lattices   clatp0cl 28799
20.3.7.6  The extended nonnegative real numbers commutative monoid   xrge0base 28813
20.3.8  Algebra   abliso 28824
20.3.8.1  Monoids Homomorphisms   abliso 28824
20.3.8.2  Ordered monoids and groups   comnd 28825
20.3.8.3  Signum in an ordered monoid   csgns 28853
20.3.8.4  The Archimedean property for generic ordered algebraic structures   cinftm 28858
20.3.8.5  Semiring left modules   cslmd 28881
20.3.8.6  Finitely supported group sums - misc additions   gsumle 28907
20.3.8.7  Rings - misc additions   rngurd 28916
20.3.8.8  Ordered rings and fields   corng 28923
20.3.8.9  Ring homomorphisms - misc additions   rhmdvdsr 28946
20.3.8.10  Scalar restriction operation   cresv 28952
20.3.8.11  The commutative ring of gaussian integers   gzcrng 28967
20.3.8.12  The archimedean ordered field of real numbers   reofld 28968
20.3.9  Matrices   symgfcoeu 28973
20.3.9.1  The symmetric group   symgfcoeu 28973
20.3.9.2  Permutation Signs   psgndmfi 28974
20.3.9.3  Submatrices   csmat 28984
20.3.9.4  Matrix literals   clmat 29002
20.3.9.5  Laplace expansion of determinants   mdetpmtr1 29014
20.3.10  Topology   fvproj 29024
20.3.10.1  Open maps   fvproj 29024
20.3.10.2  Topology of the unit circle   qtopt1 29027
20.3.10.3  Refinements   reff 29031
20.3.10.4  Open cover refinement property   ccref 29034
20.3.10.5  Lindelöf spaces   cldlf 29044
20.3.10.6  Paracompact spaces   cpcmp 29047
20.3.10.7  Pseudometrics   cmetid 29054
20.3.10.8  Continuity - misc additions   hauseqcn 29066
20.3.10.9  Topology of the closed unit   unitsscn 29067
20.3.10.10  Topology of ` ( RR X. RR ) `   unicls 29074
20.3.10.11  Order topology - misc. additions   cnvordtrestixx 29084
20.3.10.12  Continuity in topological spaces - misc. additions   mndpluscn 29097
20.3.10.13  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 29103
20.3.10.14  Limits - misc additions   lmlim 29118
20.3.10.15  Univariate polynomials   pl1cn 29126
20.3.11  Uniform Stuctures and Spaces   chcmp 29127
20.3.11.1  Hausdorff uniform completion   chcmp 29127
20.3.12  Topology and algebraic structures   zringnm 29129
20.3.12.1  The norm on the ring of the integer numbers   zringnm 29129
20.3.12.2  Topological ` ZZ ` -modules   zlm0 29131
20.3.12.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 29141
20.3.12.4  Canonical embedding of the real numbers into a complete ordered field   crrh 29162
20.3.12.5  Embedding from the extended real numbers into a complete lattice   cxrh 29185
20.3.12.6  Canonical embeddings into the ordered field of the real numbers   zrhre 29188
*20.3.12.7  Topological Manifolds   cmntop 29191
20.3.13  Real and complex functions   nexple 29196
20.3.13.1  Integer powers - misc. additions   nexple 29196
20.3.13.2  Indicator Functions   cind 29197
20.3.13.3  Extended sum   cesum 29213
20.3.14  Mixed Function/Constant operation   cofc 29281
20.3.15  Abstract measure   csiga 29294
20.3.15.1  Sigma-Algebra   csiga 29294
20.3.15.2  Generated sigma-Algebra   csigagen 29325
*20.3.15.3  lambda and pi-Systems, Rings of Sets   ispisys 29339
20.3.15.4  The Borel algebra on the real numbers   cbrsiga 29368
20.3.15.5  Product Sigma-Algebra   csx 29375
20.3.15.6  Measures   cmeas 29382
20.3.15.7  The counting measure   cntmeas 29413
20.3.15.8  The Lebesgue measure - misc additions   voliune 29416
20.3.15.9  The Dirac delta measure   cdde 29419
20.3.15.10  The 'almost everywhere' relation   cae 29424
20.3.15.11  Measurable functions   cmbfm 29436
20.3.15.12  Borel Algebra on ` ( RR X. RR ) `   br2base 29455
*20.3.15.13  Caratheodory's extension theorem   coms 29477
20.3.16  Integration   itgeq12dv 29523
20.3.16.1  Lebesgue integral - misc additions   itgeq12dv 29523
20.3.16.2  Bochner integral   citgm 29524
20.3.17  Euler's partition theorem   oddpwdc 29551
20.3.18  Sequences defined by strong recursion   csseq 29580
20.3.19  Fibonacci Numbers   cfib 29593
20.3.20  Probability   cprb 29604
20.3.20.1  Probability Theory   cprb 29604
20.3.20.2  Conditional Probabilities   ccprob 29628
20.3.20.3  Real Valued Random Variables   crrv 29637
20.3.20.4  Preimage set mapping operator   corvc 29652
20.3.20.5  Distribution Functions   orvcelval 29665
20.3.20.6  Cumulative Distribution Functions   orvclteel 29669
20.3.20.7  Probabilities - example   coinfliplem 29675
20.3.20.8  Bertrand's Ballot Problem   ballotlemoex 29682
20.3.21  Signum (sgn or sign) function - misc. additions   sgncl 29773
20.3.22  Words over a set - misc additions   wrdres 29789
20.3.22.1  Operations on words   ccatmulgnn0dir 29791
20.3.23  Polynomials with real coefficients - misc additions   plymul02 29795
20.3.24  Descartes's rule of signs   signspval 29801
20.3.24.1  Sign changes in a word over real numbers   signspval 29801
20.3.24.2  Counting sign changes in a word over real numbers   signslema 29811
20.3.25  Elementary Geometry   cstrkg2d 29841
*20.3.25.1  Two-dimension geometry   cstrkg2d 29841
20.3.25.2  Outer Five Segment (not used, no need to move to main)   cafs 29846
*20.4  Mathbox for Jonathan Ben-Naim
20.4.1  First-order logic and set theory   bnj170 29863
20.4.2  Well founded induction and recursion   bnj110 30028
20.4.3  The existence of a minimal element in certain classes   bnj69 30178
20.4.4  Well-founded induction   bnj1204 30180
20.4.5  Well-founded recursion, part 1 of 3   bnj60 30230
20.4.6  Well-founded recursion, part 2 of 3   bnj1500 30236
20.4.7  Well-founded recursion, part 3 of 3   bnj1522 30240
20.5  Mathbox for Mario Carneiro
20.5.1  Predicate calculus with all distinct variables   ax-7d 30241
20.5.2  Miscellaneous stuff   quartfull 30247
20.5.3  Derangements and the Subfactorial   deranglem 30248
20.5.4  The Erdős-Szekeres theorem   erdszelem1 30273
20.5.5  The Kuratowski closure-complement theorem   kur14lem1 30288
20.5.6  Retracts and sections   cretr 30299
20.5.7  Path-connected and simply connected spaces   cpcon 30301
20.5.8  Covering maps   ccvm 30337
20.5.9  Normal numbers   snmlff 30411
20.5.10  Godel-sets of formulas   cgoe 30415
20.5.11  Models of ZF   cgze 30443
*20.5.12  Metamath formal systems   cmcn 30457
20.5.13  Grammatical formal systems   cm0s 30582
20.5.14  Models of formal systems   cmuv 30600
20.5.15  Splitting fields   citr 30622
20.5.16  p-adic number fields   czr 30638
*20.6  Mathbox for Filip Cernatescu
20.7  Mathbox for Paul Chapman
20.7.1  Real and complex numbers (cont.)   climuzcnv 30663
20.7.2  Miscellaneous theorems   elfzm12 30667
20.8  Mathbox for Scott Fenton
20.8.1  ZFC Axioms in primitive form   axextprim 30676
20.8.2  Untangled classes   untelirr 30683
20.8.3  Extra propositional calculus theorems   3orel1 30690
20.8.4  Misc. Useful Theorems   nepss 30698
20.8.5  Properties of real and complex numbers   sqdivzi 30707
20.8.6  Infinite products   iprodefisumlem 30722
20.8.7  Factorial limits   faclimlem1 30725
20.8.8  Greatest common divisor and divisibility   pdivsq 30731
20.8.9  Properties of relationships   brtp 30735
20.8.10  Properties of functions and mappings   funpsstri 30752
20.8.11  Epsilon induction   setinds 30770
20.8.12  Ordinal numbers   elpotr 30773
20.8.13  Defined equality axioms   axextdfeq 30790
20.8.14  Hypothesis builders   hbntg 30798
20.8.15  (Trans)finite Recursion Theorems   tfisg 30803
20.8.16  Transitive closure under a relationship   ctrpred 30804
20.8.17  Founded Induction   frmin 30826
20.8.18  Ordering Ordinal Sequences   orderseqlem 30836
20.8.19  Well-founded zero, successor, and limits   cwsuc 30839
20.8.20  Founded Recursion   frr3g 30859
20.8.21  Surreal Numbers   csur 30873
20.8.22  Surreal Numbers: Ordering   sltsolem1 30903
20.8.23  Surreal Numbers: Birthday Function   bdayfo 30910
20.8.24  Surreal Numbers: Density   fvnobday 30917
20.8.25  Surreal Numbers: Upper and Lower Bounds   nobndlem1 30927
20.8.26  Surreal Numbers: Full-Eta Property   nofulllem1 30937
20.8.27  Quantifier-free definitions   ctxp 30942
20.8.28  Alternate ordered pairs   caltop 31069
20.8.29  Geometry in the Euclidean space   cofs 31095
20.8.29.1  Congruence properties   cofs 31095
20.8.29.2  Betweenness properties   btwntriv2 31125
20.8.29.3  Segment Transportation   ctransport 31142
20.8.29.4  Properties relating betweenness and congruence   cifs 31148
20.8.29.5  Connectivity of betweenness   btwnconn1lem1 31200
20.8.29.6  Segment less than or equal to   csegle 31219
20.8.29.7  Outside of relationship   coutsideof 31232
20.8.29.8  Lines and Rays   cline2 31247
20.8.30  Forward difference   cfwddif 31271
20.8.31  Rank theorems   rankung 31279
20.8.32  Hereditarily Finite Sets   chf 31285
20.9  Mathbox for Jeff Hankins
20.9.1  Miscellany   a1i14 31300
20.9.2  Basic topological facts   topbnd 31324
20.9.3  Topology of the real numbers   ivthALT 31335
20.9.4  Refinements   cfne 31336
20.9.5  Neighborhood bases determine topologies   neibastop1 31359
20.9.6  Lattice structure of topologies   topmtcl 31363
20.9.7  Filter bases   fgmin 31370
20.9.8  Directed sets, nets   tailfval 31372
20.10  Mathbox for Anthony Hart
20.10.1  Propositional Calculus   tb-ax1 31383
20.10.2  Predicate Calculus   allt 31405
20.10.3  Misc. Single Axiom Systems   meran1 31415
20.10.4  Connective Symmetry   negsym1 31421
20.11  Mathbox for Chen-Pang He
20.11.1  Ordinal topology   ontopbas 31432
20.12  Mathbox for Jeff Hoffman
20.12.1  Inferences for finite induction on generic function values   fveleq 31455
20.12.2  gdc.mm   nnssi2 31459
20.13  Mathbox for Asger C. Ipsen
20.13.1  Continuous nowhere differentiable functions   dnival 31466
*20.14  Mathbox for BJ
*20.14.1  Propositional calculus   bj-mp2c 31536
*20.14.1.1  Derived rules of inference   bj-mp2c 31536
*20.14.1.2  A syntactic theorem   bj-0 31538
20.14.1.3  Minimal implicational calculus   bj-a1k 31540
20.14.1.4  Positive calculus   bj-orim2 31546
20.14.1.5  Implication and negation   pm4.81ALT 31551
*20.14.1.6  Disjunction   bj-jaoi1 31561
*20.14.1.7  Logical equivalence   bj-dfbi4 31563
20.14.1.8  The conditional operator for propositions   bj-consensus 31567
*20.14.1.9  Propositional calculus: miscellaneous   sylancl2 31572
*20.14.2  Modal logic   bj-axdd2 31584
*20.14.3  Provability logic   cprvb 31590
*20.14.4  First-order logic   wnff 31599
20.14.4.1  Universal and existential quantifiers, "non-free" predicate   wnff 31599
20.14.4.5  Equality and substitution   wssb 31643
20.14.4.8  Membership predicate, ax-8 and ax-9   bj-elequ2g 31688
*20.14.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 31739
*20.14.4.13  Strengthenings of theorems of the main part   bj-sb3b 31834
*20.14.4.14  Distinct var metavariables   bj-hbaeb2 31835
*20.14.4.15  Around ~ equsal   bj-equsal1t 31839
*20.14.4.16  Some Principia Mathematica proofs   stdpc5t 31844
20.14.4.17  Alternate definition of substitution   bj-sbsb 31854
20.14.4.18  Lemmas for substitution   bj-sbf3 31856
20.14.4.19  Existential uniqueness   bj-eu3f 31859
*20.14.4.20  First-logic: miscellaneous   bj-nfdiOLD 31861
20.14.5  Set theory   eliminable1 31865
*20.14.5.1  Eliminability of class terms   eliminable1 31865
*20.14.5.2  Classes without extensionality   bj-eleq1w 31872
*20.14.5.3  The class-form not-free predicate   bj-nfcsym 31911
*20.14.5.4  Proposal for the definitions of class membership and class equality   bj-ax8 31912
*20.14.5.5  Lemmas for class substitution   bj-sbeqALT 31919
20.14.5.6  Removing some dv conditions   bj-exlimmpi 31929
*20.14.5.7  Class abstractions   bj-unrab 31946
*20.14.5.8  Restricted non-freeness   wrnf 31954
*20.14.5.10  Some disjointness results   bj-n0i 31959
*20.14.5.11  Complements on direct products   bj-xpimasn 31967
*20.14.5.12  "Singletonization" and tagging   bj-sels 31975
*20.14.5.13  Tuples of classes   bj-cproj 32003
*20.14.5.14  Set theory: miscellaneous   bj-vjust2 32038
*20.14.5.15  Elementwise intersection (families of sets induced on a subset)   bj-rest00 32047
20.14.5.16  Topology (complements)   bj-toptopon2 32066
20.14.5.17  Maps-to notation for functions with three arguments   bj-0nelmpt 32082
*20.14.5.18  Currying   cfset 32088
*20.14.6  Extended real and complex numbers, real and complex projectives lines   bj-elid 32094
*20.14.6.1  Diagonal in a Cartesian square   bj-elid 32094
*20.14.6.2  Extended numbers and projective lines as sets   cinftyexpi 32102
*20.14.6.4  Argument, multiplication and inverse   cprcpal 32137
*20.14.7  Monoids   bj-cmnssmnd 32145
*20.14.7.1  Finite sums in monoids   cfinsum 32154
*20.14.8  Affine, Euclidean, and Cartesian geometry   crrvec 32157
*20.14.8.1  Convex hull in real vector spaces   crrvec 32157
*20.14.8.2  Complex numbers (supplements)   bj-subcom 32163
*20.14.8.3  Barycentric coordinates   bj-bary1lem 32169
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 32252
20.17.2  Implication chains   wl-section-impchain 32276
20.17.3  An alternative definition of df-nf   wl-section-nf 32294
20.17.4  An alternative axiom ~ ax-13   ax-wl-13v 32339
20.17.5  Other stuff   wl-jarri 32341
20.18  Mathbox for Brendan Leahy
20.19.1  Logic and set theory   anim12da 32551
20.19.2  Real and complex numbers; integers   filbcmb 32580
20.19.3  Sequences and sums   sdclem2 32583
20.19.4  Topology   subspopn 32593
20.19.5  Metric spaces   metf1o 32596
20.19.6  Continuous maps and homeomorphisms   constcncf 32603
20.19.7  Boundedness   ctotbnd 32610
20.19.8  Isometries   cismty 32642
20.19.9  Heine-Borel Theorem   heibor1lem 32653
20.19.10  Banach Fixed Point Theorem   bfplem1 32666
20.19.11  Euclidean space   crrn 32669
20.19.12  Intervals (continued)   ismrer1 32682
20.19.13  Operation properties   cass 32686
20.19.14  Groups and related structures   cmagm 32692
20.19.15  Group homomorphism and isomorphism   cghomOLD 32727
20.19.16  Rings   crngo 32738
20.19.17  Division Rings   cdrng 32792
20.19.18  Ring homomorphisms   crnghom 32804
20.19.19  Commutative rings   ccm2 32833
20.19.20  Ideals   cidl 32851
20.19.21  Prime rings and integral domains   cprrng 32890
20.19.22  Ideal generators   cigen 32903
20.20  Mathbox for Giovanni Mascellani
*20.20.1  Tools for automatic proof building   efald2 32922
*20.20.2  Tseitin axioms   fald 32981
*20.20.3  Equality deductions   iuneq2f 33008
*20.20.4  Miscellanea   scottexf 33021
20.21  Mathbox for Rodolfo Medina
20.21.1  Partitions   prtlem60 33027
*20.22  Mathbox for Norm Megill
*20.22.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 33061
*20.22.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 33071
*20.22.3  Legacy theorems using obsolete axioms   ax5ALT 33085
20.22.4  Experiments with weak deduction theorem   elimhyps 33140
20.22.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 33154
20.22.7  Functionals and kernels of a left vector space (or module)   clfn 33237
20.22.8  Opposite rings and dual vector spaces   cld 33303
20.22.9  Ortholattices and orthomodular lattices   cops 33352
20.22.10  Atomic lattices with covering property   ccvr 33442
20.22.11  Hilbert lattices   chlt 33530
20.22.12  Projective geometries based on Hilbert lattices   clln 33670
20.22.13  Construction of a vector space from a Hilbert lattice   cdlema1N 33970
20.22.14  Construction of involution and inner product from a Hilbert lattice   clpoN 35662
20.23  Mathbox for OpenAI
20.24  Mathbox for Stefan O'Rear
20.24.1  Additional elementary logic and set theory   moxfr 36148
20.24.2  Additional theory of functions   imaiinfv 36149
20.24.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 36154
20.24.5  Algebraic closure systems   cnacs 36158
20.24.6  Miscellanea 1. Map utilities   constmap 36169
20.24.7  Miscellanea for polynomials   mptfcl 36176
20.24.8  Multivariate polynomials over the integers   cmzpcl 36177
20.24.9  Miscellanea for Diophantine sets 1   coeq0i 36209
20.24.10  Diophantine sets 1: definitions   cdioph 36211
20.24.11  Diophantine sets 2 miscellanea   ellz1 36223
20.24.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 36229
20.24.13  Diophantine sets 3: construction   diophrex 36232
20.24.14  Diophantine sets 4 miscellanea   2sbcrex 36241
20.24.15  Diophantine sets 4: Quantification   rexrabdioph 36251
20.24.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 36258
20.24.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 36268
20.24.18  Pigeonhole Principle and cardinality helpers   fphpd 36273
20.24.19  A non-closed set of reals is infinite   rencldnfilem 36277
20.24.20  Lagrange's rational approximation theorem   irrapxlem1 36279
20.24.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 36286
20.24.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 36293
20.24.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 36335
*20.24.24  Logarithm laws generalized to an arbitrary base   reglogcl 36347
20.24.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 36355
20.24.26  X and Y sequences 1: Definition and recurrence laws   crmx 36357
20.24.27  Ordering and induction lemmas for the integers   monotuz 36399
20.24.28  X and Y sequences 2: Order properties   rmxypos 36407
20.24.29  Congruential equations   congtr 36425
20.24.30  Alternating congruential equations   acongid 36435
20.24.31  Additional theorems on integer divisibility   coprmdvdsb 36445
20.24.32  X and Y sequences 3: Divisibility properties   jm2.18 36448
20.24.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 36465
20.24.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 36475
20.24.35  Uncategorized stuff not associated with a major project   setindtr 36484
20.24.36  More equivalents of the Axiom of Choice   axac10 36493
20.24.37  Finitely generated left modules   clfig 36530
20.24.38  Noetherian left modules I   clnm 36538
20.24.39  Addenda for structure powers   pwssplit4 36552
20.24.40  Every set admits a group structure iff choice   unxpwdom3 36558
20.24.41  Noetherian rings and left modules II   clnr 36573
20.24.42  Hilbert's Basis Theorem   cldgis 36585
20.24.43  Additional material on polynomials [DEPRECATED]   cmnc 36595
20.24.44  Degree and minimal polynomial of algebraic numbers   cdgraa 36604
20.24.45  Algebraic integers I   citgo 36628
20.24.46  Endomorphism algebra   cmend 36646
20.24.47  Subfields   csdrg 36666
20.24.48  Cyclic groups and order   idomrootle 36674
20.24.49  Cyclotomic polynomials   ccytp 36681
20.24.50  Miscellaneous topology   fgraphopab 36689
20.25  Mathbox for Jon Pennant
20.26  Mathbox for Richard Penner
20.26.1  Short Studies   ifpan123g 36704
20.26.1.1  Additional work on conditional logical operator   ifpan123g 36704
20.26.1.2  Sophisms   rp-fakeimass 36758
*20.26.1.3  Finite Sets   rp-isfinite5 36764
20.26.1.4  Infinite Sets   pwelg 36766
*20.26.1.5  Finite intersection property   fipjust 36771
20.26.1.6  RP ADDTO: Subclasses and subsets   rababg 36780
20.26.1.7  RP ADDTO: The intersection of a class   elintabg 36781
20.26.1.8  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 36784
20.26.1.9  RP ADDTO: Relations   xpinintabd 36787
*20.26.1.10  RP ADDTO: Functions   elmapintab 36803
*20.26.1.11  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 36807
20.26.1.12  RP ADDTO: First and second members of an ordered pair   elcnvlem 36808
20.26.1.13  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 36811
20.26.1.14  RP ADDTO: Basic properties of closures   cleq2lem 36815
20.26.1.15  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 36838
20.26.2  Additional statements on relations and subclasses   al3im 36839
20.26.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 36858
20.26.2.2  Reflexive closures   crcl 36865
*20.26.2.3  Finite relationship composition.   relexp2 36870
20.26.2.4  Transitive closure of a relation   dftrcl3 36913
*20.26.2.5  Adapted from Frege   frege77d 36939
*20.26.3  Propositions from _Begriffsschrift_   dfxor4 36959
*20.26.3.1  _Begriffsschrift_ Chapter I   dfxor4 36959
*20.26.3.2  _Begriffsschrift_ Notation hints   rp-imass 36967
20.26.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 36986
20.26.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 37025
*20.26.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 37052
20.26.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 37083
20.26.3.7  _Begriffsschrift_ Chapter II with equivalence of classes (where they are sets)   frege53c 37110
*20.26.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 37128
*20.26.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 37135
*20.26.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 37158
*20.26.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 37174
*20.26.4  Exploring Topology via Seifert And Threlfall   enrelmap 37193
*20.26.4.1  Equinumerosity of sets of relations and maps   enrelmap 37193
*20.26.4.2  Generic Pseudoclosure Spaces, Pseudointeror Spaces, and Pseudoneighborhoods   sscon34b 37219
*20.26.4.3  Generic Neighborhood Spaces   gneispa 37330
*20.26.5  Exploring Higher Homotopy via Kerodon   k0004lem1 37347
*20.26.5.1  Simplicial Sets   k0004lem1 37347
20.27  Mathbox for Stanislas Polu
20.27.1  IMO Problems   wwlemuld 37356
20.27.1.1  IMO 1972 B2   wwlemuld 37356
*20.27.2  INT Inequalities Proof Generator   int-addcomd 37380
20.27.4  AM-GM (for k = 2,3,4)   gsumws3 37403
20.28  Mathbox for Steve Rodriguez
20.28.1  Miscellanea   nanorxor 37408
20.28.2  Ratio test for infinite series convergence and divergence   dvgrat 37415
20.28.3  Multiples   reldvds 37418
20.28.4  Function operations   caofcan 37426
20.28.5  Calculus   lhe4.4ex1a 37432
20.28.6  The generalized binomial coefficient operation   cbcc 37439
20.28.7  Binomial series   uzmptshftfval 37449
20.29  Mathbox for Andrew Salmon
20.29.1  Principia Mathematica * 10   pm10.12 37461
20.29.2  Principia Mathematica * 11   2alanimi 37475
20.29.3  Predicate Calculus   sbeqal1 37502
20.29.4  Principia Mathematica * 13 and * 14   pm13.13a 37512
20.29.5  Set Theory   elnev 37543
20.29.7  Geometry   cplusr 37564
*20.30  Mathbox for Alan Sare
20.30.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 37586
20.30.2  Supplementary unification deductions   bi1imp 37590
20.30.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 37610
20.30.4  What is Virtual Deduction?   wvd1 37688
20.30.5  Virtual Deduction Theorems   df-vd1 37689
20.30.6  Theorems proved using Virtual Deduction   trsspwALT 37949
20.30.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 37985
20.30.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 38053
20.30.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 38057
20.30.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 38064
*20.30.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 38067
20.31  Mathbox for Glauco Siliprandi
20.31.1  Miscellanea   fnvinran 38078
20.31.2  Functions   unima 38223
20.31.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 38311
20.31.4  Real intervals   gtnelioc 38445
20.31.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 38533
20.31.7  Limits   clim1fr1 38554
20.31.8  Trigonometry   coseq0 38634
20.31.9  Continuous Functions   mulcncff 38640
20.31.10  Derivatives   dvsinexp 38685
20.31.11  Integrals   volioo 38730
20.31.12  Stone Weierstrass theorem - real version   stoweidlem1 38784
20.31.13  Wallis' product for π   wallispilem1 38848
20.31.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 38857
20.31.15  Dirichlet kernel   dirkerval 38874
20.31.16  Fourier Series   fourierdlem1 38891
20.31.17  e is transcendental   elaa2lem 39018
20.31.18  n-dimensional Euclidean space   rrxtopn 39071
20.31.19  Basic measure theory   csalg 39098
*20.31.19.1  σ-Algebras   csalg 39098
20.31.19.2  Sum of nonnegative extended reals   csumge0 39149
*20.31.19.3  Measures   cmea 39236
*20.31.19.4  Outer measures and Caratheodory's construction   come 39273
*20.31.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 39320
*20.31.19.6  Measurable functions   csmblfn 39480
20.32  Mathbox for Saveliy Skresanov
20.32.1  Ceva's theorem   sigarval 39582
20.33  Mathbox for Jarvin Udandy
20.34  Mathbox for Alexander van der Vekens
20.34.1  Double restricted existential uniqueness   r19.32 39710
20.34.1.1  Restricted quantification (extension)   r19.32 39710
20.34.1.2  The empty set (extension)   raaan2 39718
20.34.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 39719
20.34.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 39724
*20.34.2  Alternative definitions of function's and operation's values   wdfat 39736
20.34.2.1  Restricted quantification (extension)   ralbinrald 39742
20.34.2.2  The universal class (extension)   nvelim 39743
20.34.2.3  Introduce the Axiom of Power Sets (extension)   alneu 39744
20.34.2.4  Relations (extension)   eldmressn 39746
20.34.2.5  Functions (extension)   fveqvfvv 39747
20.34.2.6  Predicate "defined at"   dfateq12d 39753
20.34.2.7  Alternative definition of the value of a function   dfafv2 39756
20.34.2.8  Alternative definition of the value of an operation   aoveq123d 39802
20.34.3  General auxiliary theorems   1t10e1p1e11 39832
20.34.3.1  Miscellanea   1t10e1p1e11 39832
20.34.3.2  The modulo (remainder) operation (extension)   m1mod0mod1 39844
*20.34.3.3  Partitions of real intervals   ciccp 39846
20.34.4  Number theory (extension)   cfmtno 39872
*20.34.4.1  Fermat numbers   cfmtno 39872
*20.34.4.2  Mersenne primes   m2prm 39938
20.34.4.3  Proth's theorem   modexp2m1d 39962
*20.34.5  Even and odd numbers   ceven 39970
20.34.5.1  Definitions and basic properties   ceven 39970
20.34.5.2  Alternate definitions using the "divides" relation   dfeven2 39995
20.34.5.3  Alternate definitions using the "modulo" operation   dfeven3 40003
20.34.5.4  Alternate definitions using the "gcd" operation   iseven5 40009
20.34.5.5  Theorems of part 5 revised   zneoALTV 40013
20.34.5.6  Theorems of part 6 revised   odd2np1ALTV 40018
20.34.5.7  Theorems of AV's mathbox revised   0evenALTV 40032
20.34.5.9  Perfect Number Theorem (revised)   perfectALTVlem1 40059
*20.34.5.10  Goldbach's conjectures   cgbe 40062
20.34.6  Words over a set (extension)   wrdred1 40135
20.34.6.1  Truncated words   wrdred1 40135
20.34.6.2  Last symbol of a word (extension)   lswn0 40137
20.34.6.3  Concatenations with singleton words (extension)   ccatw2s1cl 40138
*20.34.6.4  Prefixes of a word   cpfx 40139
*20.34.7  Auxiliary theorems for graph theory   elnelall 40197
20.34.7.1  Negated equality and membership - extension   elnelall 40197
20.34.7.2  Subclasses and subsets - extension   clel5 40198
20.34.7.3  The empty set - extension   ralnralall 40202
20.34.7.4  Unordered and ordered pairs - extension   elpwdifsn 40207
20.34.7.5  Indexed union and intersection - extension   iunopeqop 40221
20.34.7.6  Ordered-pair class abstractions - extension   opabn1stprc 40223
20.34.7.7  Relations - extension   resresdm 40224
20.34.7.8  Functions - extension   fvifeq 40227
20.34.7.9  Restricted iota - extension   riotaeqimp 40255
20.34.7.10  Equinumerosity - extension   resfnfinfin 40256
20.34.7.11  Subtraction - extension   cnambpcma 40258
20.34.7.12  Ordering on reals (cont.) - extension   leaddsuble 40260
20.34.7.13  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 40266
20.34.7.14  Upper sets of integers - extension   eluzge0nn0 40267
20.34.7.15  Finite intervals of integers - extension   ssfz12 40268
20.34.7.16  Half-open integer ranges - extension   subsubelfzo0 40276
20.34.7.17  The ` # ` (set size) function - extension   nfile 40286
*20.34.7.18  Extended nonnegative integers   cxnn0 40289
20.34.7.19  Finite and infinite sums - extension   fsummsndifre 40311
20.34.8  Graph theory (revised)   cedgf 40315
20.34.8.1  The edge function extractor for extensible structures   cedgf 40315
*20.34.8.2  Vertices and edges   cvtx 40321
20.34.8.3  Undirected hypergraphs   cuhgr 40370
20.34.8.4  Undirected pseudographs and multigraphs   cupgr 40398
*20.34.8.5  Loop-free graphs   umgrislfupgrlem 40439
20.34.8.6  Edges as subsets of vertices of graphs   cedga 40443
*20.34.8.7  Undirected simple graphs - basics   cuspgr 40470
20.34.8.8  Examples for graphs   usgr0e 40554
20.34.8.9  Subgraphs   csubgr 40583
20.34.8.10  Undirected simple graphs - finite graphs   cfusgr 40627
20.34.8.11  Neighbors, complete graphs and universal vertices   cnbgr 40642
*20.34.8.12  Vertex degree   cvtxdg 40773
*20.34.8.13  Regular graphs   crgr 40847
*20.34.8.14  Walks   cewlks 40887
20.34.8.15  Walks for loop-free graphs   lfgrwlkprop 40988
20.34.8.16  Trails   ctrls 40991
20.34.8.17  Paths   cpths 41011
20.34.8.18  Closed walks   cclwlks 41068
20.34.8.19  Circuits and cycles   ccrcts 41082
*20.34.8.20  Walks as words   cwwlks 41120
20.34.8.21  Walks/paths of length 2 (as length 3 strings)   21wlkdlem1 41224
20.34.8.22  Walks in regular graphs   rusgrnumwwlkl1 41264
*20.34.8.23  Closed walks as words   cclwwlks 41275
20.34.8.24  Examples for walks, trails and paths   0ewlk 41374
20.34.8.25  Connected graphs   cconngr 41445
*20.34.8.26  Eulerian paths   ceupth 41456
*20.34.8.27  The Königsberg Bridge problem   konigsbergvtx 41506
20.34.8.28  Friendship graphs - basics   cfrgr 41520
20.34.8.29  The friendship theorem for small graphs   frgr1v 41533
20.34.8.30  Theorems according to Mertzios and Unger   2pthfrgrrn 41544
*20.34.8.31  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 41561
20.34.9  Monoids (extension)   ovn0dmfun 41646
20.34.9.1  Auxiliary theorems   ovn0dmfun 41646
20.34.9.2  Magmas and Semigroups (extension)   plusfreseq 41654
20.34.9.3  Magma homomorphisms and submagmas   cmgmhm 41659
20.34.9.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpt2ismgm 41689
*20.34.10  Magmas and internal binary operations (alternate approach)   ccllaw 41701
*20.34.10.1  Laws for internal binary operations   ccllaw 41701
*20.34.10.2  Internal binary operations   cintop 41714
20.34.10.3  Alternative definitions for Magmas and Semigroups   cmgm2 41733
20.34.11  Categories (extension)   idfusubc0 41747
20.34.11.1  Subcategories (extension)   idfusubc0 41747
20.34.12  Rings (extension)   lmod0rng 41750
20.34.12.1  Nonzero rings (extension)   lmod0rng 41750
*20.34.12.2  Non-unital rings ("rngs")   crng 41756
20.34.12.3  Rng homomorphisms   crngh 41767
20.34.12.4  Ring homomorphisms (extension)   rhmfn 41800
20.34.12.5  Ideals as non-unital rings   lidldomn1 41803
20.34.12.6  The non-unital ring of even integers   0even 41813
20.34.12.7  A constructed not unital ring   plusgndxnmulrndx 41835
*20.34.12.8  The category of non-unital rings   crngc 41841
*20.34.12.9  The category of (unital) rings   cringc 41887
20.34.12.10  Subcategories of the category of rings   srhmsubclem1 41957
20.34.13  Basic algebraic structures (extension)   xpprsng 41995
20.34.13.1  Auxiliary theorems   xpprsng 41995
20.34.13.2  The binomial coefficient operation (extension)   bcpascm1 42014
20.34.13.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 42017
20.34.13.4  Ordered group sum operation (extension)   gsumpr 42024
20.34.13.5  Symmetric groups (extension)   nn0le2is012 42030
20.34.13.6  Divisibility (extension)   invginvrid 42034
20.34.13.7  The support of functions (extension)   rmsupp0 42035
20.34.13.8  Finitely supported functions (extension)   rmsuppfi 42040
20.34.13.9  Left modules (extension)   lmodvsmdi 42049
20.34.13.10  Associative algebras (extension)   ascl0 42051
20.34.13.11  Univariate polynomials (extension)   ply1vr1smo 42055
20.34.13.12  Univariate polynomials (examples)   linply1 42067
20.34.14  Linear algebra (extension)   cdmatalt 42071
*20.34.14.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 42071
*20.34.14.2  Linear combinations   clinc 42079
*20.34.14.3  Linear independency   clininds 42115
20.34.14.4  Simple left modules and the ` ZZ `-module   lmod1lem1 42162
20.34.14.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 42182
20.34.15  Complexity theory   offval0 42185
20.34.15.1  Auxiliary theorems   offval0 42185
20.34.15.2  The modulo (remainder) operation (extension)   fldivmod 42199
20.34.15.3  Even and odd integers   nn0onn0ex 42204
20.34.15.4  The natural logarithm on complex numbers (extension)   logge0b 42215
20.34.15.5  Division of functions   cfdiv 42221
20.34.15.6  Upper bounds   cbigo 42231
20.34.15.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 42242
*20.34.15.8  The binary logarithm   fldivexpfllog2 42249
20.34.15.9  Binary length   cblen 42253
*20.34.15.10  Digits   cdig 42279
20.34.15.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 42299
20.34.15.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 42308
*20.35  Mathbox for David A. Wheeler
*20.35.2  Greater than, greater than or equal to.   cge-real 42313
*20.35.3  Hyperbolic trigonometric functions   csinh 42323
*20.35.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 42334
*20.35.5  Identities for "if"   ifnmfalse 42356
*20.35.6  Decimal point   cdp2 42357
*20.35.7  Logarithms generalized to arbitrary base using ` logb `   logb2aval 42367
*20.35.8  Logarithm laws generalized to an arbitrary base - log_   clog- 42368
*20.35.9  Formally define terms such as Reflexivity   wreflexive 42370