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
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 3888
            2.1.19  The intersection of a class   cint 3923
            2.1.20  Indexed union and intersection   ciun 3965
            2.1.21  Disjointness   wdisj 4059
            2.1.22  Binary relations   wbr 4083
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4144
            2.1.24  Transitive classes   wtr 4182
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4199
            2.2.2  Introduce the Axiom of Separation   ax-sep 4202
            2.2.3  Derive the Null Set Axiom   zfnuleu 4208
            2.2.4  Theorems requiring subset and intersection existence   nalset 4214
            2.2.5  Theorems requiring empty set existence   class2seteq 4247
            2.2.6  Collection principle   bnd 4256
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4258
            2.3.2  A notation for excluded middle   wem 4278
            2.3.3  Axiom of Pairing   ax-pr 4293
            2.3.4  Ordered pair theorem   opm 4320
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4344
            2.3.6  Power class of union and intersection   pwin 4373
            2.3.7  Epsilon and identity relations   cep 4378
            ​*2.3.8  Partial and total orderings   wpo 4385
            2.3.9  Founded and set-like relations   wfrfor 4418
            2.3.10  Ordinals   word 4453
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4524
            2.4.2  Ordinals (continued)   ordon 4578
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4624
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4629
            2.5.3  Transfinite induction   tfi 4674
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4680
            2.6.2  The natural numbers   com 4682
            2.6.3  Peano's postulates   peano1 4686
            2.6.4  Finite induction (for finite ordinals)   find 4691
            2.6.5  The Natural Numbers (continued)   nn0suc 4696
            2.6.6  Relations   cxp 4717
            2.6.7  Definite description binder (inverted iota)   cio 5276
            2.6.8  Functions   wfun 5312
            2.6.9  Cantor's Theorem   canth 5958
            2.6.10  Restricted iota (description binder)   crio 5959
            2.6.11  Operations   co 6007
            2.6.12  Maps-to notation   elmpocl 6206
            2.6.13  Function operation   cof 6222
            2.6.14  Functions (continued)   resfunexgALT 6259
            2.6.15  First and second members of an ordered pair   c1st 6290
            ​*2.6.16  Special maps-to operations   opeliunxp2f 6390
            2.6.17  Function transposition   ctpos 6396
            2.6.18  Undefined values   pwuninel2 6434
            2.6.19  Functions on ordinals; strictly monotone ordinal functions   iunon 6436
            2.6.20  "Strong" transfinite recursion   crecs 6456
            2.6.21  Recursive definition generator   crdg 6521
            2.6.22  Finite recursion   cfrec 6542
            2.6.23  Ordinal arithmetic   c1o 6561
            2.6.24  Natural number arithmetic   nna0 6628
            2.6.25  Equivalence relations and classes   wer 6685
            2.6.26  The mapping operation   cmap 6803
            2.6.27  Infinite Cartesian products   cixp 6853
            2.6.28  Equinumerosity   cen 6893
            2.6.29  Equinumerosity (cont.)   xpf1o 7013
            2.6.30  Pigeonhole Principle   phplem1 7021
            2.6.31  Finite sets   fict 7038
            2.6.32  Schroeder-Bernstein Theorem   sbthlem1 7132
            2.6.33  Finite intersections   cfi 7143
            2.6.34  Supremum and infimum   csup 7157
            2.6.35  Ordinal isomorphism   ordiso2 7210
            2.6.36  Disjoint union   cdju 7212
                  2.6.36.1  Disjoint union   cdju 7212
                  ​*2.6.36.2  Left and right injections of a disjoint union   cinl 7220
                  2.6.36.3  Universal property of the disjoint union   djuss 7245
                  2.6.36.4  Dominance and equinumerosity properties of disjoint union   djudom 7268
                  2.6.36.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7277
                  2.6.36.6  Countable sets   0ct 7282
            ​*2.6.37  The one-point compactification of the natural numbers   xnninf 7294
            2.6.38  Omniscient sets   comni 7309
            2.6.39  Markov's principle   cmarkov 7326
            2.6.40  Weakly omniscient sets   cwomni 7338
            2.6.41  Cardinal numbers   ccrd 7357
            2.6.42  Axiom of Choice equivalents   wac 7395
            2.6.43  Cardinal number arithmetic   endjudisj 7400
            2.6.44  Ordinal trichotomy   exmidontriimlem1 7411
            2.6.45  Excluded middle and the power set of a singleton   iftrueb01 7416
            2.6.46  Apartness relations   wap 7441
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7456
​​*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 7467
            4.1.2  Final derivation of real and complex number postulates   axcnex 8054
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 8098
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 8131
            4.2.2  Infinity and the extended real number system   cpnf 8186
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8221
            4.2.4  Ordering on reals   lttr 8228
            4.2.5  Initial properties of the complex numbers   mul12 8283
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8312
            4.3.2  Subtraction   cmin 8325
            4.3.3  Multiplication   kcnktkm1cn 8537
            4.3.4  Ordering on reals (cont.)   ltadd2 8574
            4.3.5  Real Apartness   creap 8729
            4.3.6  Complex Apartness   cap 8736
            4.3.7  Reciprocals   recextlem1 8806
            4.3.8  Division   cdiv 8827
            4.3.9  Ordering on reals (cont.)   ltp1 8999
            4.3.10  Suprema   lbreu 9100
            4.3.11  Imaginary and complex number properties   crap0 9113
            4.3.12  Function operation analogue theorems   ofnegsub 9117
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 9118
            4.4.2  Principle of mathematical induction   nnind 9134
            ​*4.4.3  Decimal representation of numbers   c2 9169
            ​*4.4.4  Some properties of specific numbers   neg1cn 9223
            4.4.5  Simple number properties   halfcl 9345
            4.4.6  The Archimedean property   arch 9374
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9377
            ​*4.4.8  Extended nonnegative integers   cxnn0 9440
            4.4.9  Integers (as a subset of complex numbers)   cz 9454
            4.4.10  Decimal arithmetic   cdc 9586
            4.4.11  Upper sets of integers   cuz 9730
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9822
            4.4.13  Complex numbers as pairs of reals   cnref1o 9854
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9857
            4.5.2  Infinity and the extended real number system (cont.)   cxne 9973
            4.5.3  Real number intervals   cioo 10092
            4.5.4  Finite intervals of integers   cfz 10212
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10316
            4.5.6  Half-open integer ranges   cfzo 10346
            4.5.7  Rational numbers (cont.)   qtri3or 10468
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10496
            4.6.2  The modulo (remainder) operation   cmo 10552
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10629
            4.6.4  Strong induction over upper sets of integers   uzsinds 10674
            4.6.5  The infinite sequence builder "seq"   cseq 10677
            4.6.6  Integer powers   cexp 10768
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10949
            4.6.8  Factorial function   cfa 10955
            4.6.9  The binomial coefficient operation   cbc 10977
            4.6.10  The ` # ` (set size) function   chash 11005
                  4.6.10.1  Proper unordered pairs and triples (sets of size 2 and 3)   hash2en 11073
                  4.6.10.2  Functions with a domain containing at least two different elements   fundm2domnop0 11075
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 11079
            4.7.2  Last symbol of a word   clsw 11124
            4.7.3  Concatenations of words   cconcat 11133
            4.7.4  Singleton words   cs1 11156
            4.7.5  Concatenations with singleton words   ccatws1cl 11173
            4.7.6  Subwords/substrings   csubstr 11185
            4.7.7  Prefixes of a word   cpfx 11212
            4.7.8  Subwords of subwords   swrdswrdlem 11244
            4.7.9  Subwords and concatenations   pfxcctswrd 11250
            4.7.10  Subwords of concatenations   swrdccatfn 11264
            4.7.11  Longer string literals   cs2 11289
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 11333
            4.8.2  Real and imaginary parts; conjugate   ccj 11358
            4.8.3  Sequence convergence   caucvgrelemrec 11498
            4.8.4  Square root; absolute value   csqrt 11515
            4.8.5  The maximum of two real numbers   maxcom 11722
            4.8.6  The minimum of two real numbers   mincom 11748
            4.8.7  The maximum of two extended reals   xrmaxleim 11763
            4.8.8  The minimum of two extended reals   xrnegiso 11781
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 11797
            4.9.2  Finite and infinite sums   csu 11872
            4.9.3  The binomial theorem   binomlem 12002
            4.9.4  Infinite sums (cont.)   isumshft 12009
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 12016
            4.9.6  Arithmetic series   arisum 12017
            4.9.7  Geometric series   expcnvap0 12021
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 12042
            4.9.9  Mertens' theorem   mertenslemub 12053
            4.9.10  Finite and infinite products   prodf 12057
                  4.9.10.1  Product sequences   prodf 12057
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 12067
                  4.9.10.3  Complex products   cprod 12069
                  4.9.10.4  Finite products   fprodseq 12102
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 12161
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 12294
            4.10.2  _e is irrational   eirraplem 12296
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 12306
            ​*5.1.2  Even and odd numbers   evenelz 12386
            5.1.3  The division algorithm   divalglemnn 12437
            5.1.4  Bit sequences   cbits 12459
            5.1.5  The greatest common divisor operator   cgcd 12482
            5.1.6  Bézout's identity   bezoutlemnewy 12525
            5.1.7  Decidable sets of integers   nnmindc 12563
            5.1.8  Algorithms   nn0seqcvgd 12571
            5.1.9  Euclid's Algorithm   eucalgval2 12583
            ​*5.1.10  The least common multiple   clcm 12590
            ​*5.1.11  Coprimality and Euclid's lemma   coprmgcdb 12618
            5.1.12  Cancellability of congruences   congr 12630
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12637
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12674
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12691
            5.2.4  Properties of the canonical representation of a rational   cnumer 12711
            5.2.5  Euler's theorem   codz 12738
            5.2.6  Arithmetic modulo a prime number   modprm1div 12778
            5.2.7  Pythagorean Triples   coprimeprodsq 12788
            5.2.8  The prime count function   cpc 12815
            5.2.9  Pocklington's theorem   prmpwdvds 12886
            5.2.10  Infinite primes theorem   infpnlem1 12890
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 12894
            5.2.12  Lagrange's four-square theorem   cgz 12900
            5.2.13  Decimal arithmetic (cont.)   dec2dvds 12942
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 12971
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 13036
            6.1.2  Slot definitions   cplusg 13118
            6.1.3  Definition of the structure product   crest 13280
            6.1.4  Definition of the structure quotient   cimas 13340
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 13394
            ​*7.1.2  Identity elements   mgmidmo 13413
            ​*7.1.3  Iterated sums in a magma   fngsum 13429
            ​*7.1.4  Semigroups   csgrp 13442
            ​*7.1.5  Definition and basic properties of monoids   cmnd 13457
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13498
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13534
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13541
            ​*7.2.2  Group multiple operation   cmg 13664
            7.2.3  Subgroups and Quotient groups   csubg 13712
            7.2.4  Elementary theory of group homomorphisms   cghm 13785
            7.2.5  Abelian groups   ccmn 13829
                  7.2.5.1  Definition and basic properties   ccmn 13829
                  7.2.5.2  Group sum operation   gsumfzreidx 13882
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 13891
            ​*7.3.2  Non-unital rings ("rngs")   crng 13903
            ​*7.3.3  Ring unity (multiplicative identity)   cur 13930
            7.3.4  Semirings   csrg 13934
            7.3.5  Definition and basic properties of unital rings   crg 13967
            7.3.6  Opposite ring   coppr 14038
            7.3.7  Divisibility   cdsr 14057
            7.3.8  Ring homomorphisms   crh 14122
            7.3.9  Nonzero rings and zero rings   cnzr 14151
            7.3.10  Local rings   clring 14162
            7.3.11  Subrings   csubrng 14169
                  7.3.11.1  Subrings of non-unital rings   csubrng 14169
                  7.3.11.2  Subrings of unital rings   csubrg 14189
            7.3.12  Left regular elements and domains   crlreg 14227
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 14252
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 14259
            7.5.2  Subspaces and spans in a left module   clss 14324
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 14405
            7.6.2  Ideals and spans   clidl 14439
            7.6.3  Two-sided ideals and quotient rings   c2idl 14471
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14506
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14507
            ​*7.7.2  Ring of integers   czring 14562
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14583
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Abstract multivariate polynomials
            8.1.1  Definition and basic properties   cmps 14633
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 14679
                  9.1.1.1  Topologies   ctop 14679
                  9.1.1.2  Topologies on sets   ctopon 14692
                  9.1.1.3  Topological spaces   ctps 14712
            9.1.2  Topological bases   ctb 14724
            9.1.3  Examples of topologies   distop 14767
            9.1.4  Closure and interior   ccld 14774
            9.1.5  Neighborhoods   cnei 14820
            9.1.6  Subspace topologies   restrcl 14849
            9.1.7  Limits and continuity in topological spaces   ccn 14867
            9.1.8  Product topologies   ctx 14934
            9.1.9  Continuous function-builders   cnmptid 14963
            9.1.10  Homeomorphisms   chmeo 14982
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 15004
            9.2.2  Basic metric space properties   cxms 15018
            9.2.3  Metric space balls   blfvalps 15067
            9.2.4  Open sets of a metric space   mopnrel 15123
            9.2.5  Continuity in metric spaces   metcnp3 15193
            9.2.6  Topology on the reals   qtopbasss 15203
            9.2.7  Topological definitions using the reals   ccncf 15252
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 15299
            10.1.2  Intermediate value theorem   ivthinclemlm 15316
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 15336
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 15336
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 15410
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 15450
            11.2.2  Properties of pi = 3.14159...   pilem1 15461
            11.2.3  The natural logarithm on complex numbers   clog 15538
            ​*11.2.4  Logarithms to an arbitrary base   clogb 15625
            11.2.5  Quartic binomial expansion   binom4 15661
      ​11.3  Basic number theory
            11.3.1  Wilson's theorem   wilthlem1 15662
            11.3.2  Number-theoretical functions   csgm 15663
            11.3.3  Perfect Number Theorem   mersenne 15679
            ​*11.3.4  Quadratic residues and the Legendre symbol   clgs 15684
            ​*11.3.5  Gauss' Lemma   gausslemma2dlem0a 15736
            11.3.6  Quadratic reciprocity   lgseisenlem1 15757
            11.3.7  All primes 4n+1 are the sum of two squares   2sqlem1 15801
​PART 12  GRAPH THEORY
      ​12.1  Vertices and edges
            12.1.1  The edge function extractor for extensible structures   cedgf 15813
            12.1.2  Vertices and indexed edges   cvtx 15821
                  12.1.2.1  Definitions and basic properties   cvtx 15821
                  12.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 15830
                  12.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdm2domval 15838
                  12.1.2.4  Degenerated cases of representations of graphs   vtxval0 15862
            12.1.3  Edges as range of the edge function   cedg 15866
      ​12.2  Undirected graphs
            12.2.1  Undirected hypergraphs   cuhgr 15875
            12.2.2  Undirected pseudographs and multigraphs   cupgr 15899
            ​*12.2.3  Loop-free graphs   umgrislfupgrenlem 15936
            12.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 15940
            ​*12.2.5  Undirected simple graphs   cuspgr 15959
      ​12.3  Walks, paths and cycles
            12.3.1  Walks   cwlks 16038
            12.3.2  Trails   ctrls 16099
​PART 13  GUIDES AND MISCELLANEA
      ​13.1  Guides (conventions, explanations, and examples)
            ​*13.1.1  Conventions   conventions 16109
            13.1.2  Definitional examples   ex-or 16110
​PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​14.1  Mathboxes for user contributions
            14.1.1  Mathbox guidelines   mathbox 16120
      ​14.2  Mathbox for BJ
            14.2.1  Propositional calculus   bj-nnsn 16121
                  ​*14.2.1.1  Stable formulas   bj-trst 16127
                  14.2.1.2  Decidable formulas   bj-trdc 16140
            14.2.2  Predicate calculus   bj-ex 16150
            14.2.3  Set theorey miscellaneous   bj-el2oss1o 16162
            ​*14.2.4  Extensionality   bj-vtoclgft 16163
            ​*14.2.5  Decidability of classes   wdcin 16181
            14.2.6  Disjoint union   djucllem 16188
            14.2.7  Miscellaneous   funmptd 16191
            ​*14.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 16199
                  *14.2.8.1  Bounded formulas   wbd 16199
                  ​*14.2.8.2  Bounded classes   wbdc 16227
            ​*14.2.9  CZF: Bounded separation   ax-bdsep 16271
                  14.2.9.1  Delta_0-classical logic   ax-bj-d0cl 16311
                  14.2.9.2  Inductive classes and the class of natural number ordinals   wind 16313
                  ​*14.2.9.3  The first three Peano postulates   bj-peano2 16326
            ​*14.2.10  CZF: Infinity   ax-infvn 16328
                  *14.2.10.1  The set of natural number ordinals   ax-infvn 16328
                  ​*14.2.10.2  Peano's fifth postulate   bdpeano5 16330
                  ​*14.2.10.3  Bounded induction and Peano's fourth postulate   findset 16332
            ​*14.2.11  CZF: Set induction   setindft 16352
                  *14.2.11.1  Set induction   setindft 16352
                  ​*14.2.11.2  Full induction   bj-findis 16366
            ​*14.2.12  CZF: Strong collection   ax-strcoll 16369
            ​*14.2.13  CZF: Subset collection   ax-sscoll 16374
            14.2.14  Real numbers   ax-ddkcomp 16376
      ​14.3  Mathbox for Jim Kingdon
            14.3.1  Propositional and predicate logic   nnnotnotr 16377
            14.3.2  The sizes of sets   1dom1el 16378
            14.3.3  The power set of a singleton   pwtrufal 16392
            14.3.4  Omniscience of NN+oo   0nninf 16400
            14.3.5  Schroeder-Bernstein Theorem   exmidsbthrlem 16420
            14.3.6  Real and complex numbers   qdencn 16425
            ​*14.3.7  Analytic omniscience principles   trilpolemclim 16434
            14.3.8  Supremum and infimum   supfz 16469
            14.3.9  Circle constant   taupi 16471
      ​14.4  Mathbox for Mykola Mostovenko
      ​14.5  Mathbox for David A. Wheeler
            14.5.1  Testable propositions   dftest 16473
            ​*14.5.2  Allsome quantifier   walsi 16474