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 Matthew House
      14.3  Mathbox for BJ
      14.4  Mathbox for Jim Kingdon
      14.5  Mathbox for Mykola Mostovenko
      14.6  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 5969
            2.6.10  Restricted iota (description binder)   crio 5970
            2.6.11  Operations   co 6018
            2.6.12  Maps-to notation   elmpocl 6217
            2.6.13  Function operation   cof 6233
            2.6.14  Functions (continued)   resfunexgALT 6270
            2.6.15  First and second members of an ordered pair   c1st 6301
            ​*2.6.16  Special maps-to operations   opeliunxp2f 6404
            2.6.17  Function transposition   ctpos 6410
            2.6.18  Undefined values   pwuninel2 6448
            2.6.19  Functions on ordinals; strictly monotone ordinal functions   iunon 6450
            2.6.20  "Strong" transfinite recursion   crecs 6470
            2.6.21  Recursive definition generator   crdg 6535
            2.6.22  Finite recursion   cfrec 6556
            2.6.23  Ordinal arithmetic   c1o 6575
            2.6.24  Natural number arithmetic   nna0 6642
            2.6.25  Equivalence relations and classes   wer 6699
            2.6.26  The mapping operation   cmap 6817
            2.6.27  Infinite Cartesian products   cixp 6867
            2.6.28  Equinumerosity   cen 6907
            2.6.29  Equinumerosity (cont.)   xpf1o 7030
            2.6.30  Pigeonhole Principle   phplem1 7038
            2.6.31  Finite sets   fict 7055
            2.6.32  Schroeder-Bernstein Theorem   sbthlem1 7156
            2.6.33  Finite intersections   cfi 7167
            2.6.34  Supremum and infimum   csup 7181
            2.6.35  Ordinal isomorphism   ordiso2 7234
            2.6.36  Disjoint union   cdju 7236
                  2.6.36.1  Disjoint union   cdju 7236
                  ​*2.6.36.2  Left and right injections of a disjoint union   cinl 7244
                  2.6.36.3  Universal property of the disjoint union   djuss 7269
                  2.6.36.4  Dominance and equinumerosity properties of disjoint union   djudom 7292
                  2.6.36.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7301
                  2.6.36.6  Countable sets   0ct 7306
            ​*2.6.37  The one-point compactification of the natural numbers   xnninf 7318
            2.6.38  Omniscient sets   comni 7333
            2.6.39  Markov's principle   cmarkov 7350
            2.6.40  Weakly omniscient sets   cwomni 7362
            2.6.41  Cardinal numbers   ccrd 7381
            2.6.42  Axiom of Choice equivalents   wac 7420
            2.6.43  Cardinal number arithmetic   endjudisj 7425
            2.6.44  Ordinal trichotomy   exmidontriimlem1 7436
            2.6.45  Excluded middle and the power set of a singleton   iftrueb01 7441
            2.6.46  Apartness relations   wap 7466
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7481
​​*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 7492
            4.1.2  Final derivation of real and complex number postulates   axcnex 8079
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 8123
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 8156
            4.2.2  Infinity and the extended real number system   cpnf 8211
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8246
            4.2.4  Ordering on reals   lttr 8253
            4.2.5  Initial properties of the complex numbers   mul12 8308
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8337
            4.3.2  Subtraction   cmin 8350
            4.3.3  Multiplication   kcnktkm1cn 8562
            4.3.4  Ordering on reals (cont.)   ltadd2 8599
            4.3.5  Real Apartness   creap 8754
            4.3.6  Complex Apartness   cap 8761
            4.3.7  Reciprocals   recextlem1 8831
            4.3.8  Division   cdiv 8852
            4.3.9  Ordering on reals (cont.)   ltp1 9024
            4.3.10  Suprema   lbreu 9125
            4.3.11  Imaginary and complex number properties   crap0 9138
            4.3.12  Function operation analogue theorems   ofnegsub 9142
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 9143
            4.4.2  Principle of mathematical induction   nnind 9159
            ​*4.4.3  Decimal representation of numbers   c2 9194
            ​*4.4.4  Some properties of specific numbers   neg1cn 9248
            4.4.5  Simple number properties   halfcl 9370
            4.4.6  The Archimedean property   arch 9399
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9402
            ​*4.4.8  Extended nonnegative integers   cxnn0 9465
            4.4.9  Integers (as a subset of complex numbers)   cz 9479
            4.4.10  Decimal arithmetic   cdc 9611
            4.4.11  Upper sets of integers   cuz 9755
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9853
            4.4.13  Complex numbers as pairs of reals   cnref1o 9885
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9888
            4.5.2  Infinity and the extended real number system (cont.)   cxne 10004
            4.5.3  Real number intervals   cioo 10123
            4.5.4  Finite intervals of integers   cfz 10243
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10347
            4.5.6  Half-open integer ranges   cfzo 10377
            4.5.7  Rational numbers (cont.)   qtri3or 10501
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10529
            4.6.2  The modulo (remainder) operation   cmo 10585
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10662
            4.6.4  Strong induction over upper sets of integers   uzsinds 10707
            4.6.5  The infinite sequence builder "seq"   cseq 10710
            4.6.6  Integer powers   cexp 10801
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10982
            4.6.8  Factorial function   cfa 10988
            4.6.9  The binomial coefficient operation   cbc 11010
            4.6.10  The ` # ` (set size) function   chash 11038
                  4.6.10.1  Proper unordered pairs and triples (sets of size 2 and 3)   hash2en 11108
                  4.6.10.2  Functions with a domain containing at least two different elements   fundm2domnop0 11113
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 11117
            4.7.2  Last symbol of a word   clsw 11162
            4.7.3  Concatenations of words   cconcat 11171
            4.7.4  Singleton words   cs1 11196
            4.7.5  Concatenations with singleton words   ccatws1cl 11213
            4.7.6  Subwords/substrings   csubstr 11230
            4.7.7  Prefixes of a word   cpfx 11257
            4.7.8  Subwords of subwords   swrdswrdlem 11289
            4.7.9  Subwords and concatenations   pfxcctswrd 11295
            4.7.10  Subwords of concatenations   swrdccatfn 11309
            4.7.11  Longer string literals   cs2 11334
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 11392
            4.8.2  Real and imaginary parts; conjugate   ccj 11417
            4.8.3  Sequence convergence   caucvgrelemrec 11557
            4.8.4  Square root; absolute value   csqrt 11574
            4.8.5  The maximum of two real numbers   maxcom 11781
            4.8.6  The minimum of two real numbers   mincom 11807
            4.8.7  The maximum of two extended reals   xrmaxleim 11822
            4.8.8  The minimum of two extended reals   xrnegiso 11840
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 11856
            4.9.2  Finite and infinite sums   csu 11931
            4.9.3  The binomial theorem   binomlem 12062
            4.9.4  Infinite sums (cont.)   isumshft 12069
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 12076
            4.9.6  Arithmetic series   arisum 12077
            4.9.7  Geometric series   expcnvap0 12081
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 12102
            4.9.9  Mertens' theorem   mertenslemub 12113
            4.9.10  Finite and infinite products   prodf 12117
                  4.9.10.1  Product sequences   prodf 12117
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 12127
                  4.9.10.3  Complex products   cprod 12129
                  4.9.10.4  Finite products   fprodseq 12162
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 12221
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 12354
            4.10.2  _e is irrational   eirraplem 12356
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 12366
            ​*5.1.2  Even and odd numbers   evenelz 12446
            5.1.3  The division algorithm   divalglemnn 12497
            5.1.4  Bit sequences   cbits 12519
            5.1.5  The greatest common divisor operator   cgcd 12542
            5.1.6  Bézout's identity   bezoutlemnewy 12585
            5.1.7  Decidable sets of integers   nnmindc 12623
            5.1.8  Algorithms   nn0seqcvgd 12631
            5.1.9  Euclid's Algorithm   eucalgval2 12643
            ​*5.1.10  The least common multiple   clcm 12650
            ​*5.1.11  Coprimality and Euclid's lemma   coprmgcdb 12678
            5.1.12  Cancellability of congruences   congr 12690
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12697
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12734
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12751
            5.2.4  Properties of the canonical representation of a rational   cnumer 12771
            5.2.5  Euler's theorem   codz 12798
            5.2.6  Arithmetic modulo a prime number   modprm1div 12838
            5.2.7  Pythagorean Triples   coprimeprodsq 12848
            5.2.8  The prime count function   cpc 12875
            5.2.9  Pocklington's theorem   prmpwdvds 12946
            5.2.10  Infinite primes theorem   infpnlem1 12950
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 12954
            5.2.12  Lagrange's four-square theorem   cgz 12960
            5.2.13  Decimal arithmetic (cont.)   dec2dvds 13002
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 13031
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 13096
            6.1.2  Slot definitions   cplusg 13178
            6.1.3  Definition of the structure product   crest 13340
            6.1.4  Definition of the structure quotient   cimas 13400
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 13454
            ​*7.1.2  Identity elements   mgmidmo 13473
            ​*7.1.3  Iterated sums in a magma   fngsum 13489
            ​*7.1.4  Semigroups   csgrp 13502
            ​*7.1.5  Definition and basic properties of monoids   cmnd 13517
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13558
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13594
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13601
            ​*7.2.2  Group multiple operation   cmg 13724
            7.2.3  Subgroups and Quotient groups   csubg 13772
            7.2.4  Elementary theory of group homomorphisms   cghm 13845
            7.2.5  Abelian groups   ccmn 13889
                  7.2.5.1  Definition and basic properties   ccmn 13889
                  7.2.5.2  Group sum operation   gsumfzreidx 13942
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 13952
            ​*7.3.2  Non-unital rings ("rngs")   crng 13964
            ​*7.3.3  Ring unity (multiplicative identity)   cur 13991
            7.3.4  Semirings   csrg 13995
            7.3.5  Definition and basic properties of unital rings   crg 14028
            7.3.6  Opposite ring   coppr 14099
            7.3.7  Divisibility   cdsr 14118
            7.3.8  Ring homomorphisms   crh 14183
            7.3.9  Nonzero rings and zero rings   cnzr 14212
            7.3.10  Local rings   clring 14223
            7.3.11  Subrings   csubrng 14230
                  7.3.11.1  Subrings of non-unital rings   csubrng 14230
                  7.3.11.2  Subrings of unital rings   csubrg 14250
            7.3.12  Left regular elements and domains   crlreg 14288
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 14313
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 14320
            7.5.2  Subspaces and spans in a left module   clss 14385
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 14466
            7.6.2  Ideals and spans   clidl 14500
            7.6.3  Two-sided ideals and quotient rings   c2idl 14532
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14567
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14568
            ​*7.7.2  Ring of integers   czring 14623
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14644
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Abstract multivariate polynomials
            8.1.1  Definition and basic properties   cmps 14694
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 14740
                  9.1.1.1  Topologies   ctop 14740
                  9.1.1.2  Topologies on sets   ctopon 14753
                  9.1.1.3  Topological spaces   ctps 14773
            9.1.2  Topological bases   ctb 14785
            9.1.3  Examples of topologies   distop 14828
            9.1.4  Closure and interior   ccld 14835
            9.1.5  Neighborhoods   cnei 14881
            9.1.6  Subspace topologies   restrcl 14910
            9.1.7  Limits and continuity in topological spaces   ccn 14928
            9.1.8  Product topologies   ctx 14995
            9.1.9  Continuous function-builders   cnmptid 15024
            9.1.10  Homeomorphisms   chmeo 15043
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 15065
            9.2.2  Basic metric space properties   cxms 15079
            9.2.3  Metric space balls   blfvalps 15128
            9.2.4  Open sets of a metric space   mopnrel 15184
            9.2.5  Continuity in metric spaces   metcnp3 15254
            9.2.6  Topology on the reals   qtopbasss 15264
            9.2.7  Topological definitions using the reals   ccncf 15313
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 15360
            10.1.2  Intermediate value theorem   ivthinclemlm 15377
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 15397
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 15397
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 15471
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 15511
            11.2.2  Properties of pi = 3.14159...   pilem1 15522
            11.2.3  The natural logarithm on complex numbers   clog 15599
            ​*11.2.4  Logarithms to an arbitrary base   clogb 15686
            11.2.5  Quartic binomial expansion   binom4 15722
      ​11.3  Basic number theory
            11.3.1  Wilson's theorem   wilthlem1 15723
            11.3.2  Number-theoretical functions   csgm 15724
            11.3.3  Perfect Number Theorem   mersenne 15740
            ​*11.3.4  Quadratic residues and the Legendre symbol   clgs 15745
            ​*11.3.5  Gauss' Lemma   gausslemma2dlem0a 15797
            11.3.6  Quadratic reciprocity   lgseisenlem1 15818
            11.3.7  All primes 4n+1 are the sum of two squares   2sqlem1 15862
​PART 12  GRAPH THEORY
      ​12.1  Vertices and edges
            12.1.1  The edge function extractor for extensible structures   cedgf 15874
            12.1.2  Vertices and indexed edges   cvtx 15882
                  12.1.2.1  Definitions and basic properties   cvtx 15882
                  12.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 15891
                  12.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdm2domval 15899
                  12.1.2.4  Degenerated cases of representations of graphs   vtxval0 15923
            12.1.3  Edges as range of the edge function   cedg 15927
      ​12.2  Undirected graphs
            12.2.1  Undirected hypergraphs   cuhgr 15937
            12.2.2  Undirected pseudographs and multigraphs   cupgr 15961
            ​*12.2.3  Loop-free graphs   umgrislfupgrenlem 16000
            12.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 16004
            ​*12.2.5  Undirected simple graphs   cuspgr 16023
            12.2.6  Examples for graphs   usgr0e 16102
            12.2.7  Subgraphs   csubgr 16123
            12.2.8  Vertex degree   cvtxdg 16156
      ​12.3  Walks, paths and cycles
            12.3.1  Walks   cwlks 16187
            12.3.2  Trails   ctrls 16250
            12.3.3  Closed walks as words   cclwwlk 16261
                  12.3.3.1  Closed walks as words   cclwwlk 16261
                  12.3.3.2  Closed walks of a fixed length as words   cclwwlkn 16273
                  12.3.3.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 16296
      ​12.4  Eulerian paths and the Konigsberg Bridge problem
            ​*12.4.1  Eulerian paths   ceupth 16312
            ​*12.4.2  The Königsberg Bridge problem   konigsbergvtx 16352
​PART 13  GUIDES AND MISCELLANEA
      ​13.1  Guides (conventions, explanations, and examples)
            ​*13.1.1  Conventions   conventions 16364
            13.1.2  Definitional examples   ex-or 16365
​PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​14.1  Mathboxes for user contributions
            14.1.1  Mathbox guidelines   mathbox 16375
      ​14.2  Mathbox for Matthew House
      ​14.3  Mathbox for BJ
            14.3.1  Propositional calculus   bj-nnsn 16380
                  ​*14.3.1.1  Stable formulas   bj-trst 16386
                  14.3.1.2  Decidable formulas   bj-trdc 16399
            14.3.2  Predicate calculus   bj-ex 16409
            14.3.3  Set theorey miscellaneous   bj-el2oss1o 16421
            ​*14.3.4  Extensionality   bj-vtoclgft 16422
            ​*14.3.5  Decidability of classes   wdcin 16440
            14.3.6  Disjoint union   djucllem 16447
            14.3.7  Miscellaneous   funmptd 16450
            ​*14.3.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 16458
                  *14.3.8.1  Bounded formulas   wbd 16458
                  ​*14.3.8.2  Bounded classes   wbdc 16486
            ​*14.3.9  CZF: Bounded separation   ax-bdsep 16530
                  14.3.9.1  Delta_0-classical logic   ax-bj-d0cl 16570
                  14.3.9.2  Inductive classes and the class of natural number ordinals   wind 16572
                  ​*14.3.9.3  The first three Peano postulates   bj-peano2 16585
            ​*14.3.10  CZF: Infinity   ax-infvn 16587
                  *14.3.10.1  The set of natural number ordinals   ax-infvn 16587
                  ​*14.3.10.2  Peano's fifth postulate   bdpeano5 16589
                  ​*14.3.10.3  Bounded induction and Peano's fourth postulate   findset 16591
            ​*14.3.11  CZF: Set induction   setindft 16611
                  *14.3.11.1  Set induction   setindft 16611
                  ​*14.3.11.2  Full induction   bj-findis 16625
            ​*14.3.12  CZF: Strong collection   ax-strcoll 16628
            ​*14.3.13  CZF: Subset collection   ax-sscoll 16633
            14.3.14  Real numbers   ax-ddkcomp 16635
      ​14.4  Mathbox for Jim Kingdon
            14.4.1  Propositional and predicate logic   nnnotnotr 16636
            14.4.2  The sizes of sets   ss1oel2o 16637
            14.4.3  The power set of a singleton   pwtrufal 16649
            14.4.4  Omniscience of NN+oo   0nninf 16657
            14.4.5  Schroeder-Bernstein Theorem   exmidsbthrlem 16677
            14.4.6  Real and complex numbers   qdencn 16682
            ​*14.4.7  Analytic omniscience principles   trilpolemclim 16691
            14.4.8  Supremum and infimum   supfz 16727
            14.4.9  Circle constant   taupi 16729
            14.4.10  Finite group sum over unordered finite set   cgfsu 16730
      ​14.5  Mathbox for Mykola Mostovenko
      ​14.6  Mathbox for David A. Wheeler
            14.6.1  Testable propositions   dftest 16741
            ​*14.6.2  Allsome quantifier   walsi 16742