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
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 617
            1.2.6  Logical disjunction   wo 713
            1.2.7  Stable propositions   wstab 835
            1.2.8  Decidable propositions   wdc 839
            ​*1.2.9  Theorems of decidable propositions   const 857
            1.2.10  Miscellaneous theorems of propositional calculus   pm5.21nd 921
            ​*1.2.11  The conditional operator for propositions   wif 983
            1.2.12  Abbreviated conjunction and disjunction of three wff's   w3o 1001
            1.2.13  True and false constants   wal 1393
                  ​*1.2.13.1  Universal quantifier for use by df-tru   wal 1393
                  ​*1.2.13.2  Equality predicate for use by df-tru   cv 1394
                  1.2.13.3  Define the true and false constants   wtru 1396
            1.2.14  Logical 'xor'   wxo 1417
            ​*1.2.15  Truth tables: Operations on true and false constants   truantru 1443
            ​*1.2.16  Stoic logic indemonstrables (Chrysippus of Soli)   mptnan 1465
            1.2.17  Logical implication (continued)   syl6an 1476
      ​1.3  Predicate calculus mostly without distinct variables
            ​*1.3.1  Universal quantifier (continued)   ax-5 1493
            ​*1.3.2  Equality predicate (continued)   weq 1549
            1.3.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1572
            1.3.4  Introduce Axiom of Existence   ax-i9 1576
            1.3.5  Additional intuitionistic axioms   ax-ial 1580
            1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1582
            1.3.7  The existential quantifier   19.8a 1636
            1.3.8  Equality theorems without distinct variables   a9e 1742
            1.3.9  Axioms ax-10 and ax-11   ax10o 1761
            1.3.10  Substitution (without distinct variables)   wsb 1808
            1.3.11  Theorems using axiom ax-11   equs5a 1840
      ​1.4  Predicate calculus with distinct variables
            1.4.1  Derive the axiom of distinct variables ax-16   spimv 1857
            1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1868
            1.4.3  More theorems related to ax-11 and substitution   albidv 1870
            1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1904
            1.4.5  More substitution theorems   hbs1 1989
            1.4.6  Existential uniqueness   weu 2077
            ​*1.4.7  Aristotelian logic: Assertic syllogisms   barbara 2176
      ​​*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 2211
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2215
                  2.1.2.1  Elementary properties of class abstractions   eqabdv 2358
            2.1.3  Class form not-free predicate   wnfc 2359
            2.1.4  Negated equality and membership   wne 2400
                  2.1.4.1  Negated equality   wne 2400
                  2.1.4.2  Negated membership   wnel 2495
            2.1.5  Restricted quantification   wral 2508
            2.1.6  The universal class   cvv 2799
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3011
            2.1.8  Russell's Paradox   ru 3027
            2.1.9  Proper substitution of classes for sets   wsbc 3028
            2.1.10  Proper substitution of classes for sets into classes   csb 3124
            2.1.11  Define basic set operations and relations   cdif 3194
            2.1.12  Subclasses and subsets   df-ss 3210
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3314
                  2.1.13.1  The difference of two classes   dfdif3 3314
                  2.1.13.2  The union of two classes   elun 3345
                  2.1.13.3  The intersection of two classes   elin 3387
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3435
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3470
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3484
            2.1.14  The empty set   c0 3491
            2.1.15  Conditional operator   cif 3602
            2.1.16  Power classes   cpw 3649
            2.1.17  Unordered and ordered pairs   csn 3666
            2.1.18  The union of a class   cuni 3887
            2.1.19  The intersection of a class   cint 3922
            2.1.20  Indexed union and intersection   ciun 3964
            2.1.21  Disjointness   wdisj 4058
            2.1.22  Binary relations   wbr 4082
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4143
            2.1.24  Transitive classes   wtr 4181
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4198
            2.2.2  Introduce the Axiom of Separation   ax-sep 4201
            2.2.3  Derive the Null Set Axiom   zfnuleu 4207
            2.2.4  Theorems requiring subset and intersection existence   nalset 4213
            2.2.5  Theorems requiring empty set existence   class2seteq 4246
            2.2.6  Collection principle   bnd 4255
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4257
            2.3.2  A notation for excluded middle   wem 4277
            2.3.3  Axiom of Pairing   ax-pr 4292
            2.3.4  Ordered pair theorem   opm 4319
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4343
            2.3.6  Power class of union and intersection   pwin 4370
            2.3.7  Epsilon and identity relations   cep 4375
            ​*2.3.8  Partial and total orderings   wpo 4382
            2.3.9  Founded and set-like relations   wfrfor 4415
            2.3.10  Ordinals   word 4450
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4521
            2.4.2  Ordinals (continued)   ordon 4575
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4621
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4626
            2.5.3  Transfinite induction   tfi 4671
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4677
            2.6.2  The natural numbers   com 4679
            2.6.3  Peano's postulates   peano1 4683
            2.6.4  Finite induction (for finite ordinals)   find 4688
            2.6.5  The Natural Numbers (continued)   nn0suc 4693
            2.6.6  Relations   cxp 4714
            2.6.7  Definite description binder (inverted iota)   cio 5272
            2.6.8  Functions   wfun 5308
            2.6.9  Cantor's Theorem   canth 5945
            2.6.10  Restricted iota (description binder)   crio 5946
            2.6.11  Operations   co 5994
            2.6.12  Maps-to notation   elmpocl 6191
            2.6.13  Function operation   cof 6206
            2.6.14  Functions (continued)   resfunexgALT 6243
            2.6.15  First and second members of an ordered pair   c1st 6274
            ​*2.6.16  Special maps-to operations   opeliunxp2f 6374
            2.6.17  Function transposition   ctpos 6380
            2.6.18  Undefined values   pwuninel2 6418
            2.6.19  Functions on ordinals; strictly monotone ordinal functions   iunon 6420
            2.6.20  "Strong" transfinite recursion   crecs 6440
            2.6.21  Recursive definition generator   crdg 6505
            2.6.22  Finite recursion   cfrec 6526
            2.6.23  Ordinal arithmetic   c1o 6545
            2.6.24  Natural number arithmetic   nna0 6610
            2.6.25  Equivalence relations and classes   wer 6667
            2.6.26  The mapping operation   cmap 6785
            2.6.27  Infinite Cartesian products   cixp 6835
            2.6.28  Equinumerosity   cen 6875
            2.6.29  Equinumerosity (cont.)   xpf1o 6993
            2.6.30  Pigeonhole Principle   phplem1 7001
            2.6.31  Finite sets   fict 7018
            2.6.32  Schroeder-Bernstein Theorem   sbthlem1 7112
            2.6.33  Finite intersections   cfi 7123
            2.6.34  Supremum and infimum   csup 7137
            2.6.35  Ordinal isomorphism   ordiso2 7190
            2.6.36  Disjoint union   cdju 7192
                  2.6.36.1  Disjoint union   cdju 7192
                  ​*2.6.36.2  Left and right injections of a disjoint union   cinl 7200
                  2.6.36.3  Universal property of the disjoint union   djuss 7225
                  2.6.36.4  Dominance and equinumerosity properties of disjoint union   djudom 7248
                  2.6.36.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7257
                  2.6.36.6  Countable sets   0ct 7262
            ​*2.6.37  The one-point compactification of the natural numbers   xnninf 7274
            2.6.38  Omniscient sets   comni 7289
            2.6.39  Markov's principle   cmarkov 7306
            2.6.40  Weakly omniscient sets   cwomni 7318
            2.6.41  Cardinal numbers   ccrd 7337
            2.6.42  Axiom of Choice equivalents   wac 7375
            2.6.43  Cardinal number arithmetic   endjudisj 7380
            2.6.44  Ordinal trichotomy   exmidontriimlem1 7391
            2.6.45  Excluded middle and the power set of a singleton   iftrueb01 7396
            2.6.46  Apartness relations   wap 7421
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7436
​​*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 7447
            4.1.2  Final derivation of real and complex number postulates   axcnex 8034
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 8078
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 8111
            4.2.2  Infinity and the extended real number system   cpnf 8166
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8201
            4.2.4  Ordering on reals   lttr 8208
            4.2.5  Initial properties of the complex numbers   mul12 8263
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8292
            4.3.2  Subtraction   cmin 8305
            4.3.3  Multiplication   kcnktkm1cn 8517
            4.3.4  Ordering on reals (cont.)   ltadd2 8554
            4.3.5  Real Apartness   creap 8709
            4.3.6  Complex Apartness   cap 8716
            4.3.7  Reciprocals   recextlem1 8786
            4.3.8  Division   cdiv 8807
            4.3.9  Ordering on reals (cont.)   ltp1 8979
            4.3.10  Suprema   lbreu 9080
            4.3.11  Imaginary and complex number properties   crap0 9093
            4.3.12  Function operation analogue theorems   ofnegsub 9097
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 9098
            4.4.2  Principle of mathematical induction   nnind 9114
            ​*4.4.3  Decimal representation of numbers   c2 9149
            ​*4.4.4  Some properties of specific numbers   neg1cn 9203
            4.4.5  Simple number properties   halfcl 9325
            4.4.6  The Archimedean property   arch 9354
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9357
            ​*4.4.8  Extended nonnegative integers   cxnn0 9420
            4.4.9  Integers (as a subset of complex numbers)   cz 9434
            4.4.10  Decimal arithmetic   cdc 9566
            4.4.11  Upper sets of integers   cuz 9710
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9802
            4.4.13  Complex numbers as pairs of reals   cnref1o 9834
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9837
            4.5.2  Infinity and the extended real number system (cont.)   cxne 9953
            4.5.3  Real number intervals   cioo 10072
            4.5.4  Finite intervals of integers   cfz 10192
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10296
            4.5.6  Half-open integer ranges   cfzo 10326
            4.5.7  Rational numbers (cont.)   qtri3or 10447
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10475
            4.6.2  The modulo (remainder) operation   cmo 10531
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10608
            4.6.4  Strong induction over upper sets of integers   uzsinds 10653
            4.6.5  The infinite sequence builder "seq"   cseq 10656
            4.6.6  Integer powers   cexp 10747
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10928
            4.6.8  Factorial function   cfa 10934
            4.6.9  The binomial coefficient operation   cbc 10956
            4.6.10  The ` # ` (set size) function   chash 10984
                  4.6.10.1  Proper unordered pairs and triples (sets of size 2 and 3)   hash2en 11052
                  4.6.10.2  Functions with a domain containing at least two different elements   fundm2domnop0 11054
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 11058
            4.7.2  Last symbol of a word   clsw 11102
            4.7.3  Concatenations of words   cconcat 11111
            4.7.4  Singleton words   cs1 11134
            4.7.5  Concatenations with singleton words   ccatws1cl 11151
            4.7.6  Subwords/substrings   csubstr 11163
            4.7.7  Prefixes of a word   cpfx 11190
            4.7.8  Subwords of subwords   swrdswrdlem 11222
            4.7.9  Subwords and concatenations   pfxcctswrd 11228
            4.7.10  Subwords of concatenations   swrdccatfn 11242
            4.7.11  Longer string literals   cs2 11267
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 11311
            4.8.2  Real and imaginary parts; conjugate   ccj 11336
            4.8.3  Sequence convergence   caucvgrelemrec 11476
            4.8.4  Square root; absolute value   csqrt 11493
            4.8.5  The maximum of two real numbers   maxcom 11700
            4.8.6  The minimum of two real numbers   mincom 11726
            4.8.7  The maximum of two extended reals   xrmaxleim 11741
            4.8.8  The minimum of two extended reals   xrnegiso 11759
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 11775
            4.9.2  Finite and infinite sums   csu 11850
            4.9.3  The binomial theorem   binomlem 11980
            4.9.4  Infinite sums (cont.)   isumshft 11987
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 11994
            4.9.6  Arithmetic series   arisum 11995
            4.9.7  Geometric series   expcnvap0 11999
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 12020
            4.9.9  Mertens' theorem   mertenslemub 12031
            4.9.10  Finite and infinite products   prodf 12035
                  4.9.10.1  Product sequences   prodf 12035
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 12045
                  4.9.10.3  Complex products   cprod 12047
                  4.9.10.4  Finite products   fprodseq 12080
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 12139
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 12272
            4.10.2  _e is irrational   eirraplem 12274
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 12284
            ​*5.1.2  Even and odd numbers   evenelz 12364
            5.1.3  The division algorithm   divalglemnn 12415
            5.1.4  Bit sequences   cbits 12437
            5.1.5  The greatest common divisor operator   cgcd 12460
            5.1.6  Bézout's identity   bezoutlemnewy 12503
            5.1.7  Decidable sets of integers   nnmindc 12541
            5.1.8  Algorithms   nn0seqcvgd 12549
            5.1.9  Euclid's Algorithm   eucalgval2 12561
            ​*5.1.10  The least common multiple   clcm 12568
            ​*5.1.11  Coprimality and Euclid's lemma   coprmgcdb 12596
            5.1.12  Cancellability of congruences   congr 12608
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12615
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12652
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12669
            5.2.4  Properties of the canonical representation of a rational   cnumer 12689
            5.2.5  Euler's theorem   codz 12716
            5.2.6  Arithmetic modulo a prime number   modprm1div 12756
            5.2.7  Pythagorean Triples   coprimeprodsq 12766
            5.2.8  The prime count function   cpc 12793
            5.2.9  Pocklington's theorem   prmpwdvds 12864
            5.2.10  Infinite primes theorem   infpnlem1 12868
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 12872
            5.2.12  Lagrange's four-square theorem   cgz 12878
            5.2.13  Decimal arithmetic (cont.)   dec2dvds 12920
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 12949
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 13014
            6.1.2  Slot definitions   cplusg 13096
            6.1.3  Definition of the structure product   crest 13258
            6.1.4  Definition of the structure quotient   cimas 13318
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 13372
            ​*7.1.2  Identity elements   mgmidmo 13391
            ​*7.1.3  Iterated sums in a magma   fngsum 13407
            ​*7.1.4  Semigroups   csgrp 13420
            ​*7.1.5  Definition and basic properties of monoids   cmnd 13435
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13476
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13512
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13519
            ​*7.2.2  Group multiple operation   cmg 13642
            7.2.3  Subgroups and Quotient groups   csubg 13690
            7.2.4  Elementary theory of group homomorphisms   cghm 13763
            7.2.5  Abelian groups   ccmn 13807
                  7.2.5.1  Definition and basic properties   ccmn 13807
                  7.2.5.2  Group sum operation   gsumfzreidx 13860
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 13869
            ​*7.3.2  Non-unital rings ("rngs")   crng 13881
            ​*7.3.3  Ring unity (multiplicative identity)   cur 13908
            7.3.4  Semirings   csrg 13912
            7.3.5  Definition and basic properties of unital rings   crg 13945
            7.3.6  Opposite ring   coppr 14016
            7.3.7  Divisibility   cdsr 14035
            7.3.8  Ring homomorphisms   crh 14099
            7.3.9  Nonzero rings and zero rings   cnzr 14128
            7.3.10  Local rings   clring 14139
            7.3.11  Subrings   csubrng 14146
                  7.3.11.1  Subrings of non-unital rings   csubrng 14146
                  7.3.11.2  Subrings of unital rings   csubrg 14166
            7.3.12  Left regular elements and domains   crlreg 14204
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 14229
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 14236
            7.5.2  Subspaces and spans in a left module   clss 14301
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 14382
            7.6.2  Ideals and spans   clidl 14416
            7.6.3  Two-sided ideals and quotient rings   c2idl 14448
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14483
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14484
            ​*7.7.2  Ring of integers   czring 14539
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14560
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Abstract multivariate polynomials
            8.1.1  Definition and basic properties   cmps 14610
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 14656
                  9.1.1.1  Topologies   ctop 14656
                  9.1.1.2  Topologies on sets   ctopon 14669
                  9.1.1.3  Topological spaces   ctps 14689
            9.1.2  Topological bases   ctb 14701
            9.1.3  Examples of topologies   distop 14744
            9.1.4  Closure and interior   ccld 14751
            9.1.5  Neighborhoods   cnei 14797
            9.1.6  Subspace topologies   restrcl 14826
            9.1.7  Limits and continuity in topological spaces   ccn 14844
            9.1.8  Product topologies   ctx 14911
            9.1.9  Continuous function-builders   cnmptid 14940
            9.1.10  Homeomorphisms   chmeo 14959
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 14981
            9.2.2  Basic metric space properties   cxms 14995
            9.2.3  Metric space balls   blfvalps 15044
            9.2.4  Open sets of a metric space   mopnrel 15100
            9.2.5  Continuity in metric spaces   metcnp3 15170
            9.2.6  Topology on the reals   qtopbasss 15180
            9.2.7  Topological definitions using the reals   ccncf 15229
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 15276
            10.1.2  Intermediate value theorem   ivthinclemlm 15293
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 15313
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 15313
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 15387
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 15427
            11.2.2  Properties of pi = 3.14159...   pilem1 15438
            11.2.3  The natural logarithm on complex numbers   clog 15515
            ​*11.2.4  Logarithms to an arbitrary base   clogb 15602
            11.2.5  Quartic binomial expansion   binom4 15638
      ​11.3  Basic number theory
            11.3.1  Wilson's theorem   wilthlem1 15639
            11.3.2  Number-theoretical functions   csgm 15640
            11.3.3  Perfect Number Theorem   mersenne 15656
            ​*11.3.4  Quadratic residues and the Legendre symbol   clgs 15661
            ​*11.3.5  Gauss' Lemma   gausslemma2dlem0a 15713
            11.3.6  Quadratic reciprocity   lgseisenlem1 15734
            11.3.7  All primes 4n+1 are the sum of two squares   2sqlem1 15778
​PART 12  GRAPH THEORY
      ​12.1  Vertices and edges
            12.1.1  The edge function extractor for extensible structures   cedgf 15790
            12.1.2  Vertices and indexed edges   cvtx 15798
                  12.1.2.1  Definitions and basic properties   cvtx 15798
                  12.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 15807
                  12.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdm2domval 15815
                  12.1.2.4  Degenerated cases of representations of graphs   vtxval0 15839
            12.1.3  Edges as range of the edge function   cedg 15843
      ​12.2  Undirected graphs
            12.2.1  Undirected hypergraphs   cuhgr 15852
            12.2.2  Undirected pseudographs and multigraphs   cupgr 15876
            ​*12.2.3  Loop-free graphs   umgrislfupgrenlem 15913
            12.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 15917
            ​*12.2.5  Undirected simple graphs   cuspgr 15936
​PART 13  GUIDES AND MISCELLANEA
      ​13.1  Guides (conventions, explanations, and examples)
            ​*13.1.1  Conventions   conventions 16015
            13.1.2  Definitional examples   ex-or 16016
​PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​14.1  Mathboxes for user contributions
            14.1.1  Mathbox guidelines   mathbox 16026
      ​14.2  Mathbox for BJ
            14.2.1  Propositional calculus   bj-nnsn 16027
                  ​*14.2.1.1  Stable formulas   bj-trst 16033
                  14.2.1.2  Decidable formulas   bj-trdc 16046
            14.2.2  Predicate calculus   bj-ex 16056
            14.2.3  Set theorey miscellaneous   bj-el2oss1o 16068
            ​*14.2.4  Extensionality   bj-vtoclgft 16069
            ​*14.2.5  Decidability of classes   wdcin 16087
            14.2.6  Disjoint union   djucllem 16094
            14.2.7  Miscellaneous   funmptd 16097
            ​*14.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 16105
                  *14.2.8.1  Bounded formulas   wbd 16105
                  ​*14.2.8.2  Bounded classes   wbdc 16133
            ​*14.2.9  CZF: Bounded separation   ax-bdsep 16177
                  14.2.9.1  Delta_0-classical logic   ax-bj-d0cl 16217
                  14.2.9.2  Inductive classes and the class of natural number ordinals   wind 16219
                  ​*14.2.9.3  The first three Peano postulates   bj-peano2 16232
            ​*14.2.10  CZF: Infinity   ax-infvn 16234
                  *14.2.10.1  The set of natural number ordinals   ax-infvn 16234
                  ​*14.2.10.2  Peano's fifth postulate   bdpeano5 16236
                  ​*14.2.10.3  Bounded induction and Peano's fourth postulate   findset 16238
            ​*14.2.11  CZF: Set induction   setindft 16258
                  *14.2.11.1  Set induction   setindft 16258
                  ​*14.2.11.2  Full induction   bj-findis 16272
            ​*14.2.12  CZF: Strong collection   ax-strcoll 16275
            ​*14.2.13  CZF: Subset collection   ax-sscoll 16280
            14.2.14  Real numbers   ax-ddkcomp 16282
      ​14.3  Mathbox for Jim Kingdon
            14.3.1  Propositional and predicate logic   nnnotnotr 16283
            14.3.2  The sizes of sets   1dom1el 16284
            14.3.3  The power set of a singleton   pwtrufal 16294
            14.3.4  Omniscience of NN+oo   0nninf 16301
            14.3.5  Schroeder-Bernstein Theorem   exmidsbthrlem 16321
            14.3.6  Real and complex numbers   qdencn 16326
            ​*14.3.7  Analytic omniscience principles   trilpolemclim 16335
            14.3.8  Supremum and infimum   supfz 16370
            14.3.9  Circle constant   taupi 16372
      ​14.4  Mathbox for Mykola Mostovenko
      ​14.5  Mathbox for David A. Wheeler
            14.5.1  Testable propositions   dftest 16374
            ​*14.5.2  Allsome quantifier   walsi 16375