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 2800
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3012
            2.1.8  Russell's Paradox   ru 3028
            2.1.9  Proper substitution of classes for sets   wsbc 3029
            2.1.10  Proper substitution of classes for sets into classes   csb 3125
            2.1.11  Define basic set operations and relations   cdif 3195
            2.1.12  Subclasses and subsets   df-ss 3211
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3315
                  2.1.13.1  The difference of two classes   dfdif3 3315
                  2.1.13.2  The union of two classes   elun 3346
                  2.1.13.3  The intersection of two classes   elin 3388
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3436
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3471
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3485
            2.1.14  The empty set   c0 3492
            2.1.15  Conditional operator   cif 3603
            2.1.16  Power classes   cpw 3650
            2.1.17  Unordered and ordered pairs   csn 3667
            2.1.18  The union of a class   cuni 3889
            2.1.19  The intersection of a class   cint 3924
            2.1.20  Indexed union and intersection   ciun 3966
            2.1.21  Disjointness   wdisj 4060
            2.1.22  Binary relations   wbr 4084
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4145
            2.1.24  Transitive classes   wtr 4183
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4200
            2.2.2  Introduce the Axiom of Separation   ax-sep 4203
            2.2.3  Derive the Null Set Axiom   zfnuleu 4209
            2.2.4  Theorems requiring subset and intersection existence   nalset 4215
            2.2.5  Theorems requiring empty set existence   class2seteq 4249
            2.2.6  Collection principle   bnd 4258
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4260
            2.3.2  A notation for excluded middle   wem 4280
            2.3.3  Axiom of Pairing   ax-pr 4295
            2.3.4  Ordered pair theorem   opm 4322
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4346
            2.3.6  Power class of union and intersection   pwin 4375
            2.3.7  Epsilon and identity relations   cep 4380
            ​*2.3.8  Partial and total orderings   wpo 4387
            2.3.9  Founded and set-like relations   wfrfor 4420
            2.3.10  Ordinals   word 4455
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4526
            2.4.2  Ordinals (continued)   ordon 4580
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4626
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4631
            2.5.3  Transfinite induction   tfi 4676
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4682
            2.6.2  The natural numbers   com 4684
            2.6.3  Peano's postulates   peano1 4688
            2.6.4  Finite induction (for finite ordinals)   find 4693
            2.6.5  The Natural Numbers (continued)   nn0suc 4698
            2.6.6  Relations   cxp 4719
            2.6.7  Definite description binder (inverted iota)   cio 5280
            2.6.8  Functions   wfun 5316
            2.6.9  Cantor's Theorem   canth 5962
            2.6.10  Restricted iota (description binder)   crio 5963
            2.6.11  Operations   co 6011
            2.6.12  Maps-to notation   elmpocl 6210
            2.6.13  Function operation   cof 6226
            2.6.14  Functions (continued)   resfunexgALT 6263
            2.6.15  First and second members of an ordered pair   c1st 6294
            ​*2.6.16  Special maps-to operations   opeliunxp2f 6397
            2.6.17  Function transposition   ctpos 6403
            2.6.18  Undefined values   pwuninel2 6441
            2.6.19  Functions on ordinals; strictly monotone ordinal functions   iunon 6443
            2.6.20  "Strong" transfinite recursion   crecs 6463
            2.6.21  Recursive definition generator   crdg 6528
            2.6.22  Finite recursion   cfrec 6549
            2.6.23  Ordinal arithmetic   c1o 6568
            2.6.24  Natural number arithmetic   nna0 6635
            2.6.25  Equivalence relations and classes   wer 6692
            2.6.26  The mapping operation   cmap 6810
            2.6.27  Infinite Cartesian products   cixp 6860
            2.6.28  Equinumerosity   cen 6900
            2.6.29  Equinumerosity (cont.)   xpf1o 7023
            2.6.30  Pigeonhole Principle   phplem1 7031
            2.6.31  Finite sets   fict 7048
            2.6.32  Schroeder-Bernstein Theorem   sbthlem1 7145
            2.6.33  Finite intersections   cfi 7156
            2.6.34  Supremum and infimum   csup 7170
            2.6.35  Ordinal isomorphism   ordiso2 7223
            2.6.36  Disjoint union   cdju 7225
                  2.6.36.1  Disjoint union   cdju 7225
                  ​*2.6.36.2  Left and right injections of a disjoint union   cinl 7233
                  2.6.36.3  Universal property of the disjoint union   djuss 7258
                  2.6.36.4  Dominance and equinumerosity properties of disjoint union   djudom 7281
                  2.6.36.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7290
                  2.6.36.6  Countable sets   0ct 7295
            ​*2.6.37  The one-point compactification of the natural numbers   xnninf 7307
            2.6.38  Omniscient sets   comni 7322
            2.6.39  Markov's principle   cmarkov 7339
            2.6.40  Weakly omniscient sets   cwomni 7351
            2.6.41  Cardinal numbers   ccrd 7370
            2.6.42  Axiom of Choice equivalents   wac 7408
            2.6.43  Cardinal number arithmetic   endjudisj 7413
            2.6.44  Ordinal trichotomy   exmidontriimlem1 7424
            2.6.45  Excluded middle and the power set of a singleton   iftrueb01 7429
            2.6.46  Apartness relations   wap 7454
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7469
​​*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 7480
            4.1.2  Final derivation of real and complex number postulates   axcnex 8067
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 8111
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 8144
            4.2.2  Infinity and the extended real number system   cpnf 8199
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8234
            4.2.4  Ordering on reals   lttr 8241
            4.2.5  Initial properties of the complex numbers   mul12 8296
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8325
            4.3.2  Subtraction   cmin 8338
            4.3.3  Multiplication   kcnktkm1cn 8550
            4.3.4  Ordering on reals (cont.)   ltadd2 8587
            4.3.5  Real Apartness   creap 8742
            4.3.6  Complex Apartness   cap 8749
            4.3.7  Reciprocals   recextlem1 8819
            4.3.8  Division   cdiv 8840
            4.3.9  Ordering on reals (cont.)   ltp1 9012
            4.3.10  Suprema   lbreu 9113
            4.3.11  Imaginary and complex number properties   crap0 9126
            4.3.12  Function operation analogue theorems   ofnegsub 9130
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 9131
            4.4.2  Principle of mathematical induction   nnind 9147
            ​*4.4.3  Decimal representation of numbers   c2 9182
            ​*4.4.4  Some properties of specific numbers   neg1cn 9236
            4.4.5  Simple number properties   halfcl 9358
            4.4.6  The Archimedean property   arch 9387
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9390
            ​*4.4.8  Extended nonnegative integers   cxnn0 9453
            4.4.9  Integers (as a subset of complex numbers)   cz 9467
            4.4.10  Decimal arithmetic   cdc 9599
            4.4.11  Upper sets of integers   cuz 9743
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9841
            4.4.13  Complex numbers as pairs of reals   cnref1o 9873
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9876
            4.5.2  Infinity and the extended real number system (cont.)   cxne 9992
            4.5.3  Real number intervals   cioo 10111
            4.5.4  Finite intervals of integers   cfz 10231
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10335
            4.5.6  Half-open integer ranges   cfzo 10365
            4.5.7  Rational numbers (cont.)   qtri3or 10488
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10516
            4.6.2  The modulo (remainder) operation   cmo 10572
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10649
            4.6.4  Strong induction over upper sets of integers   uzsinds 10694
            4.6.5  The infinite sequence builder "seq"   cseq 10697
            4.6.6  Integer powers   cexp 10788
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10969
            4.6.8  Factorial function   cfa 10975
            4.6.9  The binomial coefficient operation   cbc 10997
            4.6.10  The ` # ` (set size) function   chash 11025
                  4.6.10.1  Proper unordered pairs and triples (sets of size 2 and 3)   hash2en 11094
                  4.6.10.2  Functions with a domain containing at least two different elements   fundm2domnop0 11096
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 11100
            4.7.2  Last symbol of a word   clsw 11145
            4.7.3  Concatenations of words   cconcat 11154
            4.7.4  Singleton words   cs1 11179
            4.7.5  Concatenations with singleton words   ccatws1cl 11196
            4.7.6  Subwords/substrings   csubstr 11213
            4.7.7  Prefixes of a word   cpfx 11240
            4.7.8  Subwords of subwords   swrdswrdlem 11272
            4.7.9  Subwords and concatenations   pfxcctswrd 11278
            4.7.10  Subwords of concatenations   swrdccatfn 11292
            4.7.11  Longer string literals   cs2 11317
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 11362
            4.8.2  Real and imaginary parts; conjugate   ccj 11387
            4.8.3  Sequence convergence   caucvgrelemrec 11527
            4.8.4  Square root; absolute value   csqrt 11544
            4.8.5  The maximum of two real numbers   maxcom 11751
            4.8.6  The minimum of two real numbers   mincom 11777
            4.8.7  The maximum of two extended reals   xrmaxleim 11792
            4.8.8  The minimum of two extended reals   xrnegiso 11810
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 11826
            4.9.2  Finite and infinite sums   csu 11901
            4.9.3  The binomial theorem   binomlem 12031
            4.9.4  Infinite sums (cont.)   isumshft 12038
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 12045
            4.9.6  Arithmetic series   arisum 12046
            4.9.7  Geometric series   expcnvap0 12050
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 12071
            4.9.9  Mertens' theorem   mertenslemub 12082
            4.9.10  Finite and infinite products   prodf 12086
                  4.9.10.1  Product sequences   prodf 12086
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 12096
                  4.9.10.3  Complex products   cprod 12098
                  4.9.10.4  Finite products   fprodseq 12131
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 12190
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 12323
            4.10.2  _e is irrational   eirraplem 12325
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 12335
            ​*5.1.2  Even and odd numbers   evenelz 12415
            5.1.3  The division algorithm   divalglemnn 12466
            5.1.4  Bit sequences   cbits 12488
            5.1.5  The greatest common divisor operator   cgcd 12511
            5.1.6  Bézout's identity   bezoutlemnewy 12554
            5.1.7  Decidable sets of integers   nnmindc 12592
            5.1.8  Algorithms   nn0seqcvgd 12600
            5.1.9  Euclid's Algorithm   eucalgval2 12612
            ​*5.1.10  The least common multiple   clcm 12619
            ​*5.1.11  Coprimality and Euclid's lemma   coprmgcdb 12647
            5.1.12  Cancellability of congruences   congr 12659
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12666
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12703
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12720
            5.2.4  Properties of the canonical representation of a rational   cnumer 12740
            5.2.5  Euler's theorem   codz 12767
            5.2.6  Arithmetic modulo a prime number   modprm1div 12807
            5.2.7  Pythagorean Triples   coprimeprodsq 12817
            5.2.8  The prime count function   cpc 12844
            5.2.9  Pocklington's theorem   prmpwdvds 12915
            5.2.10  Infinite primes theorem   infpnlem1 12919
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 12923
            5.2.12  Lagrange's four-square theorem   cgz 12929
            5.2.13  Decimal arithmetic (cont.)   dec2dvds 12971
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 13000
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 13065
            6.1.2  Slot definitions   cplusg 13147
            6.1.3  Definition of the structure product   crest 13309
            6.1.4  Definition of the structure quotient   cimas 13369
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 13423
            ​*7.1.2  Identity elements   mgmidmo 13442
            ​*7.1.3  Iterated sums in a magma   fngsum 13458
            ​*7.1.4  Semigroups   csgrp 13471
            ​*7.1.5  Definition and basic properties of monoids   cmnd 13486
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13527
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13563
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13570
            ​*7.2.2  Group multiple operation   cmg 13693
            7.2.3  Subgroups and Quotient groups   csubg 13741
            7.2.4  Elementary theory of group homomorphisms   cghm 13814
            7.2.5  Abelian groups   ccmn 13858
                  7.2.5.1  Definition and basic properties   ccmn 13858
                  7.2.5.2  Group sum operation   gsumfzreidx 13911
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 13920
            ​*7.3.2  Non-unital rings ("rngs")   crng 13932
            ​*7.3.3  Ring unity (multiplicative identity)   cur 13959
            7.3.4  Semirings   csrg 13963
            7.3.5  Definition and basic properties of unital rings   crg 13996
            7.3.6  Opposite ring   coppr 14067
            7.3.7  Divisibility   cdsr 14086
            7.3.8  Ring homomorphisms   crh 14151
            7.3.9  Nonzero rings and zero rings   cnzr 14180
            7.3.10  Local rings   clring 14191
            7.3.11  Subrings   csubrng 14198
                  7.3.11.1  Subrings of non-unital rings   csubrng 14198
                  7.3.11.2  Subrings of unital rings   csubrg 14218
            7.3.12  Left regular elements and domains   crlreg 14256
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 14281
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 14288
            7.5.2  Subspaces and spans in a left module   clss 14353
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 14434
            7.6.2  Ideals and spans   clidl 14468
            7.6.3  Two-sided ideals and quotient rings   c2idl 14500
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14535
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14536
            ​*7.7.2  Ring of integers   czring 14591
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14612
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Abstract multivariate polynomials
            8.1.1  Definition and basic properties   cmps 14662
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 14708
                  9.1.1.1  Topologies   ctop 14708
                  9.1.1.2  Topologies on sets   ctopon 14721
                  9.1.1.3  Topological spaces   ctps 14741
            9.1.2  Topological bases   ctb 14753
            9.1.3  Examples of topologies   distop 14796
            9.1.4  Closure and interior   ccld 14803
            9.1.5  Neighborhoods   cnei 14849
            9.1.6  Subspace topologies   restrcl 14878
            9.1.7  Limits and continuity in topological spaces   ccn 14896
            9.1.8  Product topologies   ctx 14963
            9.1.9  Continuous function-builders   cnmptid 14992
            9.1.10  Homeomorphisms   chmeo 15011
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 15033
            9.2.2  Basic metric space properties   cxms 15047
            9.2.3  Metric space balls   blfvalps 15096
            9.2.4  Open sets of a metric space   mopnrel 15152
            9.2.5  Continuity in metric spaces   metcnp3 15222
            9.2.6  Topology on the reals   qtopbasss 15232
            9.2.7  Topological definitions using the reals   ccncf 15281
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 15328
            10.1.2  Intermediate value theorem   ivthinclemlm 15345
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 15365
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 15365
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 15439
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 15479
            11.2.2  Properties of pi = 3.14159...   pilem1 15490
            11.2.3  The natural logarithm on complex numbers   clog 15567
            ​*11.2.4  Logarithms to an arbitrary base   clogb 15654
            11.2.5  Quartic binomial expansion   binom4 15690
      ​11.3  Basic number theory
            11.3.1  Wilson's theorem   wilthlem1 15691
            11.3.2  Number-theoretical functions   csgm 15692
            11.3.3  Perfect Number Theorem   mersenne 15708
            ​*11.3.4  Quadratic residues and the Legendre symbol   clgs 15713
            ​*11.3.5  Gauss' Lemma   gausslemma2dlem0a 15765
            11.3.6  Quadratic reciprocity   lgseisenlem1 15786
            11.3.7  All primes 4n+1 are the sum of two squares   2sqlem1 15830
​PART 12  GRAPH THEORY
      ​12.1  Vertices and edges
            12.1.1  The edge function extractor for extensible structures   cedgf 15842
            12.1.2  Vertices and indexed edges   cvtx 15850
                  12.1.2.1  Definitions and basic properties   cvtx 15850
                  12.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 15859
                  12.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdm2domval 15867
                  12.1.2.4  Degenerated cases of representations of graphs   vtxval0 15891
            12.1.3  Edges as range of the edge function   cedg 15895
      ​12.2  Undirected graphs
            12.2.1  Undirected hypergraphs   cuhgr 15904
            12.2.2  Undirected pseudographs and multigraphs   cupgr 15928
            ​*12.2.3  Loop-free graphs   umgrislfupgrenlem 15965
            12.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 15969
            ​*12.2.5  Undirected simple graphs   cuspgr 15988
            12.2.6  Examples for graphs   usgr0e 16067
            12.2.7  Vertex degree   cvtxdg 16088
      ​12.3  Walks, paths and cycles
            12.3.1  Walks   cwlks 16105
            12.3.2  Trails   ctrls 16166
            12.3.3  Closed walks as words   cclwwlk 16176
                  12.3.3.1  Closed walks as words   cclwwlk 16176
                  12.3.3.2  Closed walks of a fixed length as words   cclwwlkn 16188
                  12.3.3.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 16211
​PART 13  GUIDES AND MISCELLANEA
      ​13.1  Guides (conventions, explanations, and examples)
            ​*13.1.1  Conventions   conventions 16227
            13.1.2  Definitional examples   ex-or 16228
​PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​14.1  Mathboxes for user contributions
            14.1.1  Mathbox guidelines   mathbox 16238
      ​14.2  Mathbox for BJ
            14.2.1  Propositional calculus   bj-nnsn 16239
                  ​*14.2.1.1  Stable formulas   bj-trst 16245
                  14.2.1.2  Decidable formulas   bj-trdc 16258
            14.2.2  Predicate calculus   bj-ex 16268
            14.2.3  Set theorey miscellaneous   bj-el2oss1o 16280
            ​*14.2.4  Extensionality   bj-vtoclgft 16281
            ​*14.2.5  Decidability of classes   wdcin 16299
            14.2.6  Disjoint union   djucllem 16306
            14.2.7  Miscellaneous   funmptd 16309
            ​*14.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 16317
                  *14.2.8.1  Bounded formulas   wbd 16317
                  ​*14.2.8.2  Bounded classes   wbdc 16345
            ​*14.2.9  CZF: Bounded separation   ax-bdsep 16389
                  14.2.9.1  Delta_0-classical logic   ax-bj-d0cl 16429
                  14.2.9.2  Inductive classes and the class of natural number ordinals   wind 16431
                  ​*14.2.9.3  The first three Peano postulates   bj-peano2 16444
            ​*14.2.10  CZF: Infinity   ax-infvn 16446
                  *14.2.10.1  The set of natural number ordinals   ax-infvn 16446
                  ​*14.2.10.2  Peano's fifth postulate   bdpeano5 16448
                  ​*14.2.10.3  Bounded induction and Peano's fourth postulate   findset 16450
            ​*14.2.11  CZF: Set induction   setindft 16470
                  *14.2.11.1  Set induction   setindft 16470
                  ​*14.2.11.2  Full induction   bj-findis 16484
            ​*14.2.12  CZF: Strong collection   ax-strcoll 16487
            ​*14.2.13  CZF: Subset collection   ax-sscoll 16492
            14.2.14  Real numbers   ax-ddkcomp 16494
      ​14.3  Mathbox for Jim Kingdon
            14.3.1  Propositional and predicate logic   nnnotnotr 16495
            14.3.2  The sizes of sets   ss1oel2o 16496
            14.3.3  The power set of a singleton   pwtrufal 16508
            14.3.4  Omniscience of NN+oo   0nninf 16516
            14.3.5  Schroeder-Bernstein Theorem   exmidsbthrlem 16536
            14.3.6  Real and complex numbers   qdencn 16541
            ​*14.3.7  Analytic omniscience principles   trilpolemclim 16550
            14.3.8  Supremum and infimum   supfz 16585
            14.3.9  Circle constant   taupi 16587
      ​14.4  Mathbox for Mykola Mostovenko
      ​14.5  Mathbox for David A. Wheeler
            14.5.1  Testable propositions   dftest 16589
            ​*14.5.2  Allsome quantifier   walsi 16590