TOC of Theorem List - Intuitionistic Logic Explorer

Table of Contents Summary

PART 1  INTUITIONISTIC FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Predicate calculus mostly without distinct variables
      1.4  Predicate calculus with distinct variables
      1.5  First-order logic with one non-logical binary predicate
PART 2  SET THEORY
      2.1  IZF Set Theory - start with the Axiom of Extensionality
      2.2  IZF Set Theory - add the Axioms of Collection and Separation
      2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
      2.4  IZF Set Theory - add the Axiom of Union
      2.5  IZF Set Theory - add the Axiom of Set Induction
      2.6  IZF Set Theory - add the Axiom of Infinity
PART 3  CHOICE PRINCIPLES
      3.1  Countable Choice and Dependent Choice
PART 4  REAL AND COMPLEX NUMBERS
      4.1  Construction and axiomatization of real and complex numbers
      4.2  Derive the basic properties from the field axioms
      4.3  Real and complex numbers - basic operations
      4.4  Integer sets
      4.5  Order sets
      4.6  Elementary integer functions
      4.7  Words over a set
      4.8  Elementary real and complex functions
      4.9  Elementary limits and convergence
      4.10  Elementary trigonometry
PART 5  ELEMENTARY NUMBER THEORY
      5.1  Elementary properties of divisibility
      5.2  Elementary prime number theory
      5.3  Cardinality of real and complex number subsets
PART 6  BASIC STRUCTURES
      6.1  Extensible structures
PART 7  BASIC ALGEBRAIC STRUCTURES
      7.1  Monoids
      7.2  Groups
      7.3  Rings
      7.4  Division rings and fields
      7.5  Left modules
      7.6  Subring algebras and ideals
      7.7  The complex numbers as an algebraic extensible structure
PART 8  BASIC LINEAR ALGEBRA
      8.1  Abstract multivariate polynomials
PART 9  BASIC TOPOLOGY
      9.1  Topology
      9.2  Metric spaces
PART 10  BASIC REAL AND COMPLEX ANALYSIS
      10.1  Continuity
      10.2  Derivatives
PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      11.1  Polynomials
      11.2  Basic trigonometry
      11.3  Basic number theory
PART 12  GRAPH THEORY
      12.1  Vertices and edges
      12.2  Undirected graphs
      12.3  Walks, paths and cycles
      12.4  Eulerian paths and the Konigsberg Bridge problem
PART 13  GUIDES AND MISCELLANEA
      13.1  Guides (conventions, explanations, and examples)
PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​14.1  Mathboxes for user contributions
      14.2  Mathbox for BJ
      14.3  Mathbox for Jim Kingdon
      14.4  Mathbox for Mykola Mostovenko
      14.5  Mathbox for David A. Wheeler

Detailed Table of Contents
(* means the section header has a description)

​​*PART 1  INTUITIONISTIC FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            1.2.2  Propositional logic axioms for implication   ax-mp 5
            ​*1.2.3  Logical implication   mp2b 8
            1.2.4  Logical conjunction and logical equivalence   wa 104
            1.2.5  Logical negation (intuitionistic)   ax-in1 619
            1.2.6  Logical disjunction   wo 715
            1.2.7  Stable propositions   wstab 837
            1.2.8  Decidable propositions   wdc 841
            ​*1.2.9  Theorems of decidable propositions   const 859
            1.2.10  Miscellaneous theorems of propositional calculus   pm5.21nd 923
            ​*1.2.11  The conditional operator for propositions   wif 985
            1.2.12  Abbreviated conjunction and disjunction of three wff's   w3o 1003
            1.2.13  True and false constants   wal 1395
                  ​*1.2.13.1  Universal quantifier for use by df-tru   wal 1395
                  ​*1.2.13.2  Equality predicate for use by df-tru   cv 1396
                  1.2.13.3  Define the true and false constants   wtru 1398
            1.2.14  Logical 'xor'   wxo 1419
            ​*1.2.15  Truth tables: Operations on true and false constants   truantru 1445
            ​*1.2.16  Stoic logic indemonstrables (Chrysippus of Soli)   mptnan 1467
            1.2.17  Logical implication (continued)   syl6an 1478
      ​1.3  Predicate calculus mostly without distinct variables
            ​*1.3.1  Universal quantifier (continued)   ax-5 1495
            ​*1.3.2  Equality predicate (continued)   weq 1551
            1.3.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1574
            1.3.4  Introduce Axiom of Existence   ax-i9 1578
            1.3.5  Additional intuitionistic axioms   ax-ial 1582
            1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1584
            1.3.7  The existential quantifier   19.8a 1638
            1.3.8  Equality theorems without distinct variables   a9e 1744
            1.3.9  Axioms ax-10 and ax-11   ax10o 1763
            1.3.10  Substitution (without distinct variables)   wsb 1810
            1.3.11  Theorems using axiom ax-11   equs5a 1842
      ​1.4  Predicate calculus with distinct variables
            1.4.1  Derive the axiom of distinct variables ax-16   spimv 1859
            1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1870
            1.4.3  More theorems related to ax-11 and substitution   albidv 1872
            1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1906
            1.4.5  More substitution theorems   hbs1 1991
            1.4.6  Existential uniqueness   weu 2079
            ​*1.4.7  Aristotelian logic: Assertic syllogisms   barbara 2178
      ​​*1.5  First-order logic with one non-logical binary predicate
​​*PART 2  SET THEORY
      ​2.1  IZF Set Theory - start with the Axiom of Extensionality
            2.1.1  Introduce the Axiom of Extensionality   ax-ext 2213
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2217
                  2.1.2.1  Elementary properties of class abstractions   eqabdv 2360
            2.1.3  Class form not-free predicate   wnfc 2361
            2.1.4  Negated equality and membership   wne 2402
                  2.1.4.1  Negated equality   wne 2402
                  2.1.4.2  Negated membership   wnel 2497
            2.1.5  Restricted quantification   wral 2510
            2.1.6  The universal class   cvv 2802
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3014
            2.1.8  Russell's Paradox   ru 3030
            2.1.9  Proper substitution of classes for sets   wsbc 3031
            2.1.10  Proper substitution of classes for sets into classes   csb 3127
            2.1.11  Define basic set operations and relations   cdif 3197
            2.1.12  Subclasses and subsets   df-ss 3213
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3317
                  2.1.13.1  The difference of two classes   dfdif3 3317
                  2.1.13.2  The union of two classes   elun 3348
                  2.1.13.3  The intersection of two classes   elin 3390
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3438
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3473
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3487
            2.1.14  The empty set   c0 3494
            2.1.15  Conditional operator   cif 3605
            2.1.16  Power classes   cpw 3652
            2.1.17  Unordered and ordered pairs   csn 3669
            2.1.18  The union of a class   cuni 3893
            2.1.19  The intersection of a class   cint 3928
            2.1.20  Indexed union and intersection   ciun 3970
            2.1.21  Disjointness   wdisj 4064
            2.1.22  Binary relations   wbr 4088
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4149
            2.1.24  Transitive classes   wtr 4187
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4204
            2.2.2  Introduce the Axiom of Separation   ax-sep 4207
            2.2.3  Derive the Null Set Axiom   zfnuleu 4213
            2.2.4  Theorems requiring subset and intersection existence   nalset 4219
            2.2.5  Theorems requiring empty set existence   class2seteq 4253
            2.2.6  Collection principle   bnd 4262
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4264
            2.3.2  A notation for excluded middle   wem 4284
            2.3.3  Axiom of Pairing   ax-pr 4299
            2.3.4  Ordered pair theorem   opm 4326
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4350
            2.3.6  Power class of union and intersection   pwin 4379
            2.3.7  Epsilon and identity relations   cep 4384
            ​*2.3.8  Partial and total orderings   wpo 4391
            2.3.9  Founded and set-like relations   wfrfor 4424
            2.3.10  Ordinals   word 4459
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4530
            2.4.2  Ordinals (continued)   ordon 4584
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4630
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4635
            2.5.3  Transfinite induction   tfi 4680
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4686
            2.6.2  The natural numbers   com 4688
            2.6.3  Peano's postulates   peano1 4692
            2.6.4  Finite induction (for finite ordinals)   find 4697
            2.6.5  The Natural Numbers (continued)   nn0suc 4702
            2.6.6  Relations   cxp 4723
            2.6.7  Definite description binder (inverted iota)   cio 5284
            2.6.8  Functions   wfun 5320
            2.6.9  Cantor's Theorem   canth 5968
            2.6.10  Restricted iota (description binder)   crio 5969
            2.6.11  Operations   co 6017
            2.6.12  Maps-to notation   elmpocl 6216
            2.6.13  Function operation   cof 6232
            2.6.14  Functions (continued)   resfunexgALT 6269
            2.6.15  First and second members of an ordered pair   c1st 6300
            ​*2.6.16  Special maps-to operations   opeliunxp2f 6403
            2.6.17  Function transposition   ctpos 6409
            2.6.18  Undefined values   pwuninel2 6447
            2.6.19  Functions on ordinals; strictly monotone ordinal functions   iunon 6449
            2.6.20  "Strong" transfinite recursion   crecs 6469
            2.6.21  Recursive definition generator   crdg 6534
            2.6.22  Finite recursion   cfrec 6555
            2.6.23  Ordinal arithmetic   c1o 6574
            2.6.24  Natural number arithmetic   nna0 6641
            2.6.25  Equivalence relations and classes   wer 6698
            2.6.26  The mapping operation   cmap 6816
            2.6.27  Infinite Cartesian products   cixp 6866
            2.6.28  Equinumerosity   cen 6906
            2.6.29  Equinumerosity (cont.)   xpf1o 7029
            2.6.30  Pigeonhole Principle   phplem1 7037
            2.6.31  Finite sets   fict 7054
            2.6.32  Schroeder-Bernstein Theorem   sbthlem1 7155
            2.6.33  Finite intersections   cfi 7166
            2.6.34  Supremum and infimum   csup 7180
            2.6.35  Ordinal isomorphism   ordiso2 7233
            2.6.36  Disjoint union   cdju 7235
                  2.6.36.1  Disjoint union   cdju 7235
                  ​*2.6.36.2  Left and right injections of a disjoint union   cinl 7243
                  2.6.36.3  Universal property of the disjoint union   djuss 7268
                  2.6.36.4  Dominance and equinumerosity properties of disjoint union   djudom 7291
                  2.6.36.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7300
                  2.6.36.6  Countable sets   0ct 7305
            ​*2.6.37  The one-point compactification of the natural numbers   xnninf 7317
            2.6.38  Omniscient sets   comni 7332
            2.6.39  Markov's principle   cmarkov 7349
            2.6.40  Weakly omniscient sets   cwomni 7361
            2.6.41  Cardinal numbers   ccrd 7380
            2.6.42  Axiom of Choice equivalents   wac 7419
            2.6.43  Cardinal number arithmetic   endjudisj 7424
            2.6.44  Ordinal trichotomy   exmidontriimlem1 7435
            2.6.45  Excluded middle and the power set of a singleton   iftrueb01 7440
            2.6.46  Apartness relations   wap 7465
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7480
​​*PART 4  REAL AND COMPLEX NUMBERS
      ​4.1  Construction and axiomatization of real and complex numbers
            4.1.1  Dedekind-cut construction of real and complex numbers   cnpi 7491
            4.1.2  Final derivation of real and complex number postulates   axcnex 8078
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 8122
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 8155
            4.2.2  Infinity and the extended real number system   cpnf 8210
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8245
            4.2.4  Ordering on reals   lttr 8252
            4.2.5  Initial properties of the complex numbers   mul12 8307
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8336
            4.3.2  Subtraction   cmin 8349
            4.3.3  Multiplication   kcnktkm1cn 8561
            4.3.4  Ordering on reals (cont.)   ltadd2 8598
            4.3.5  Real Apartness   creap 8753
            4.3.6  Complex Apartness   cap 8760
            4.3.7  Reciprocals   recextlem1 8830
            4.3.8  Division   cdiv 8851
            4.3.9  Ordering on reals (cont.)   ltp1 9023
            4.3.10  Suprema   lbreu 9124
            4.3.11  Imaginary and complex number properties   crap0 9137
            4.3.12  Function operation analogue theorems   ofnegsub 9141
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 9142
            4.4.2  Principle of mathematical induction   nnind 9158
            ​*4.4.3  Decimal representation of numbers   c2 9193
            ​*4.4.4  Some properties of specific numbers   neg1cn 9247
            4.4.5  Simple number properties   halfcl 9369
            4.4.6  The Archimedean property   arch 9398
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9401
            ​*4.4.8  Extended nonnegative integers   cxnn0 9464
            4.4.9  Integers (as a subset of complex numbers)   cz 9478
            4.4.10  Decimal arithmetic   cdc 9610
            4.4.11  Upper sets of integers   cuz 9754
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9852
            4.4.13  Complex numbers as pairs of reals   cnref1o 9884
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9887
            4.5.2  Infinity and the extended real number system (cont.)   cxne 10003
            4.5.3  Real number intervals   cioo 10122
            4.5.4  Finite intervals of integers   cfz 10242
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10346
            4.5.6  Half-open integer ranges   cfzo 10376
            4.5.7  Rational numbers (cont.)   qtri3or 10499
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10527
            4.6.2  The modulo (remainder) operation   cmo 10583
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10660
            4.6.4  Strong induction over upper sets of integers   uzsinds 10705
            4.6.5  The infinite sequence builder "seq"   cseq 10708
            4.6.6  Integer powers   cexp 10799
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10980
            4.6.8  Factorial function   cfa 10986
            4.6.9  The binomial coefficient operation   cbc 11008
            4.6.10  The ` # ` (set size) function   chash 11036
                  4.6.10.1  Proper unordered pairs and triples (sets of size 2 and 3)   hash2en 11106
                  4.6.10.2  Functions with a domain containing at least two different elements   fundm2domnop0 11108
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 11112
            4.7.2  Last symbol of a word   clsw 11157
            4.7.3  Concatenations of words   cconcat 11166
            4.7.4  Singleton words   cs1 11191
            4.7.5  Concatenations with singleton words   ccatws1cl 11208
            4.7.6  Subwords/substrings   csubstr 11225
            4.7.7  Prefixes of a word   cpfx 11252
            4.7.8  Subwords of subwords   swrdswrdlem 11284
            4.7.9  Subwords and concatenations   pfxcctswrd 11290
            4.7.10  Subwords of concatenations   swrdccatfn 11304
            4.7.11  Longer string literals   cs2 11329
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 11374
            4.8.2  Real and imaginary parts; conjugate   ccj 11399
            4.8.3  Sequence convergence   caucvgrelemrec 11539
            4.8.4  Square root; absolute value   csqrt 11556
            4.8.5  The maximum of two real numbers   maxcom 11763
            4.8.6  The minimum of two real numbers   mincom 11789
            4.8.7  The maximum of two extended reals   xrmaxleim 11804
            4.8.8  The minimum of two extended reals   xrnegiso 11822
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 11838
            4.9.2  Finite and infinite sums   csu 11913
            4.9.3  The binomial theorem   binomlem 12043
            4.9.4  Infinite sums (cont.)   isumshft 12050
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 12057
            4.9.6  Arithmetic series   arisum 12058
            4.9.7  Geometric series   expcnvap0 12062
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 12083
            4.9.9  Mertens' theorem   mertenslemub 12094
            4.9.10  Finite and infinite products   prodf 12098
                  4.9.10.1  Product sequences   prodf 12098
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 12108
                  4.9.10.3  Complex products   cprod 12110
                  4.9.10.4  Finite products   fprodseq 12143
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 12202
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 12335
            4.10.2  _e is irrational   eirraplem 12337
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 12347
            ​*5.1.2  Even and odd numbers   evenelz 12427
            5.1.3  The division algorithm   divalglemnn 12478
            5.1.4  Bit sequences   cbits 12500
            5.1.5  The greatest common divisor operator   cgcd 12523
            5.1.6  Bézout's identity   bezoutlemnewy 12566
            5.1.7  Decidable sets of integers   nnmindc 12604
            5.1.8  Algorithms   nn0seqcvgd 12612
            5.1.9  Euclid's Algorithm   eucalgval2 12624
            ​*5.1.10  The least common multiple   clcm 12631
            ​*5.1.11  Coprimality and Euclid's lemma   coprmgcdb 12659
            5.1.12  Cancellability of congruences   congr 12671
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12678
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12715
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12732
            5.2.4  Properties of the canonical representation of a rational   cnumer 12752
            5.2.5  Euler's theorem   codz 12779
            5.2.6  Arithmetic modulo a prime number   modprm1div 12819
            5.2.7  Pythagorean Triples   coprimeprodsq 12829
            5.2.8  The prime count function   cpc 12856
            5.2.9  Pocklington's theorem   prmpwdvds 12927
            5.2.10  Infinite primes theorem   infpnlem1 12931
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 12935
            5.2.12  Lagrange's four-square theorem   cgz 12941
            5.2.13  Decimal arithmetic (cont.)   dec2dvds 12983
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 13012
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 13077
            6.1.2  Slot definitions   cplusg 13159
            6.1.3  Definition of the structure product   crest 13321
            6.1.4  Definition of the structure quotient   cimas 13381
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 13435
            ​*7.1.2  Identity elements   mgmidmo 13454
            ​*7.1.3  Iterated sums in a magma   fngsum 13470
            ​*7.1.4  Semigroups   csgrp 13483
            ​*7.1.5  Definition and basic properties of monoids   cmnd 13498
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13539
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13575
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13582
            ​*7.2.2  Group multiple operation   cmg 13705
            7.2.3  Subgroups and Quotient groups   csubg 13753
            7.2.4  Elementary theory of group homomorphisms   cghm 13826
            7.2.5  Abelian groups   ccmn 13870
                  7.2.5.1  Definition and basic properties   ccmn 13870
                  7.2.5.2  Group sum operation   gsumfzreidx 13923
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 13932
            ​*7.3.2  Non-unital rings ("rngs")   crng 13944
            ​*7.3.3  Ring unity (multiplicative identity)   cur 13971
            7.3.4  Semirings   csrg 13975
            7.3.5  Definition and basic properties of unital rings   crg 14008
            7.3.6  Opposite ring   coppr 14079
            7.3.7  Divisibility   cdsr 14098
            7.3.8  Ring homomorphisms   crh 14163
            7.3.9  Nonzero rings and zero rings   cnzr 14192
            7.3.10  Local rings   clring 14203
            7.3.11  Subrings   csubrng 14210
                  7.3.11.1  Subrings of non-unital rings   csubrng 14210
                  7.3.11.2  Subrings of unital rings   csubrg 14230
            7.3.12  Left regular elements and domains   crlreg 14268
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 14293
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 14300
            7.5.2  Subspaces and spans in a left module   clss 14365
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 14446
            7.6.2  Ideals and spans   clidl 14480
            7.6.3  Two-sided ideals and quotient rings   c2idl 14512
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14547
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14548
            ​*7.7.2  Ring of integers   czring 14603
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14624
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Abstract multivariate polynomials
            8.1.1  Definition and basic properties   cmps 14674
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 14720
                  9.1.1.1  Topologies   ctop 14720
                  9.1.1.2  Topologies on sets   ctopon 14733
                  9.1.1.3  Topological spaces   ctps 14753
            9.1.2  Topological bases   ctb 14765
            9.1.3  Examples of topologies   distop 14808
            9.1.4  Closure and interior   ccld 14815
            9.1.5  Neighborhoods   cnei 14861
            9.1.6  Subspace topologies   restrcl 14890
            9.1.7  Limits and continuity in topological spaces   ccn 14908
            9.1.8  Product topologies   ctx 14975
            9.1.9  Continuous function-builders   cnmptid 15004
            9.1.10  Homeomorphisms   chmeo 15023
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 15045
            9.2.2  Basic metric space properties   cxms 15059
            9.2.3  Metric space balls   blfvalps 15108
            9.2.4  Open sets of a metric space   mopnrel 15164
            9.2.5  Continuity in metric spaces   metcnp3 15234
            9.2.6  Topology on the reals   qtopbasss 15244
            9.2.7  Topological definitions using the reals   ccncf 15293
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 15340
            10.1.2  Intermediate value theorem   ivthinclemlm 15357
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 15377
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 15377
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 15451
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 15491
            11.2.2  Properties of pi = 3.14159...   pilem1 15502
            11.2.3  The natural logarithm on complex numbers   clog 15579
            ​*11.2.4  Logarithms to an arbitrary base   clogb 15666
            11.2.5  Quartic binomial expansion   binom4 15702
      ​11.3  Basic number theory
            11.3.1  Wilson's theorem   wilthlem1 15703
            11.3.2  Number-theoretical functions   csgm 15704
            11.3.3  Perfect Number Theorem   mersenne 15720
            ​*11.3.4  Quadratic residues and the Legendre symbol   clgs 15725
            ​*11.3.5  Gauss' Lemma   gausslemma2dlem0a 15777
            11.3.6  Quadratic reciprocity   lgseisenlem1 15798
            11.3.7  All primes 4n+1 are the sum of two squares   2sqlem1 15842
​PART 12  GRAPH THEORY
      ​12.1  Vertices and edges
            12.1.1  The edge function extractor for extensible structures   cedgf 15854
            12.1.2  Vertices and indexed edges   cvtx 15862
                  12.1.2.1  Definitions and basic properties   cvtx 15862
                  12.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 15871
                  12.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdm2domval 15879
                  12.1.2.4  Degenerated cases of representations of graphs   vtxval0 15903
            12.1.3  Edges as range of the edge function   cedg 15907
      ​12.2  Undirected graphs
            12.2.1  Undirected hypergraphs   cuhgr 15917
            12.2.2  Undirected pseudographs and multigraphs   cupgr 15941
            ​*12.2.3  Loop-free graphs   umgrislfupgrenlem 15980
            12.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 15984
            ​*12.2.5  Undirected simple graphs   cuspgr 16003
            12.2.6  Examples for graphs   usgr0e 16082
            12.2.7  Subgraphs   csubgr 16103
            12.2.8  Vertex degree   cvtxdg 16136
      ​12.3  Walks, paths and cycles
            12.3.1  Walks   cwlks 16167
            12.3.2  Trails   ctrls 16230
            12.3.3  Closed walks as words   cclwwlk 16241
                  12.3.3.1  Closed walks as words   cclwwlk 16241
                  12.3.3.2  Closed walks of a fixed length as words   cclwwlkn 16253
                  12.3.3.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 16276
      ​12.4  Eulerian paths and the Konigsberg Bridge problem
            ​*12.4.1  Eulerian paths   ceupth 16292
​PART 13  GUIDES AND MISCELLANEA
      ​13.1  Guides (conventions, explanations, and examples)
            ​*13.1.1  Conventions   conventions 16317
            13.1.2  Definitional examples   ex-or 16318
​PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​14.1  Mathboxes for user contributions
            14.1.1  Mathbox guidelines   mathbox 16328
      ​14.2  Mathbox for BJ
            14.2.1  Propositional calculus   bj-nnsn 16329
                  ​*14.2.1.1  Stable formulas   bj-trst 16335
                  14.2.1.2  Decidable formulas   bj-trdc 16348
            14.2.2  Predicate calculus   bj-ex 16358
            14.2.3  Set theorey miscellaneous   bj-el2oss1o 16370
            ​*14.2.4  Extensionality   bj-vtoclgft 16371
            ​*14.2.5  Decidability of classes   wdcin 16389
            14.2.6  Disjoint union   djucllem 16396
            14.2.7  Miscellaneous   funmptd 16399
            ​*14.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 16407
                  *14.2.8.1  Bounded formulas   wbd 16407
                  ​*14.2.8.2  Bounded classes   wbdc 16435
            ​*14.2.9  CZF: Bounded separation   ax-bdsep 16479
                  14.2.9.1  Delta_0-classical logic   ax-bj-d0cl 16519
                  14.2.9.2  Inductive classes and the class of natural number ordinals   wind 16521
                  ​*14.2.9.3  The first three Peano postulates   bj-peano2 16534
            ​*14.2.10  CZF: Infinity   ax-infvn 16536
                  *14.2.10.1  The set of natural number ordinals   ax-infvn 16536
                  ​*14.2.10.2  Peano's fifth postulate   bdpeano5 16538
                  ​*14.2.10.3  Bounded induction and Peano's fourth postulate   findset 16540
            ​*14.2.11  CZF: Set induction   setindft 16560
                  *14.2.11.1  Set induction   setindft 16560
                  ​*14.2.11.2  Full induction   bj-findis 16574
            ​*14.2.12  CZF: Strong collection   ax-strcoll 16577
            ​*14.2.13  CZF: Subset collection   ax-sscoll 16582
            14.2.14  Real numbers   ax-ddkcomp 16584
      ​14.3  Mathbox for Jim Kingdon
            14.3.1  Propositional and predicate logic   nnnotnotr 16585
            14.3.2  The sizes of sets   ss1oel2o 16586
            14.3.3  The power set of a singleton   pwtrufal 16598
            14.3.4  Omniscience of NN+oo   0nninf 16606
            14.3.5  Schroeder-Bernstein Theorem   exmidsbthrlem 16626
            14.3.6  Real and complex numbers   qdencn 16631
            ​*14.3.7  Analytic omniscience principles   trilpolemclim 16640
            14.3.8  Supremum and infimum   supfz 16675
            14.3.9  Circle constant   taupi 16677
            14.3.10  Finite group sum over unordered finite set   cgfsu 16678
      ​14.4  Mathbox for Mykola Mostovenko
      ​14.5  Mathbox for David A. Wheeler
            14.5.1  Testable propositions   dftest 16686
            ​*14.5.2  Allsome quantifier   walsi 16687