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  Associative algebras
      8.2  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  Pell equations
      11.4  Basic number theory
PART 12  GRAPH THEORY
      12.1  Vertices and edges
      12.2  Undirected graphs
      12.3  Walks, paths and cycles
      12.4  Eulerian paths and the Konigsberg Bridge problem
PART 13  GUIDES AND MISCELLANEA
      13.1  Guides (conventions, explanations, and examples)
PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​14.1  Mathboxes for user contributions
      14.2  Mathbox for Matthew House
      14.3  Mathbox for BJ
      14.4  Mathbox for Jim Kingdon
      14.5  Mathbox for Mykola Mostovenko
      14.6  Mathbox for David A. Wheeler

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

​​*PART 1  INTUITIONISTIC FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            1.2.2  Propositional logic axioms for implication   ax-mp 5
            ​*1.2.3  Logical implication   mp2b 8
            1.2.4  Logical conjunction and logical equivalence   wa 104
            1.2.5  Logical negation (intuitionistic)   ax-in1 623
            1.2.6  Logical disjunction   wo 720
            1.2.7  Stable propositions   wstab 842
            1.2.8  Decidable propositions   wdc 846
            ​*1.2.9  Theorems of decidable propositions   const 864
            1.2.10  Miscellaneous theorems of propositional calculus   pm5.21nd 928
            ​*1.2.11  The conditional operator for propositions   wif 990
            1.2.12  Abbreviated conjunction and disjunction of three wff's   w3o 1008
            1.2.13  True and false constants   wal 1400
                  ​*1.2.13.1  Universal quantifier for use by df-tru   wal 1400
                  ​*1.2.13.2  Equality predicate for use by df-tru   cv 1401
                  1.2.13.3  Define the true and false constants   wtru 1403
            1.2.14  Logical 'xor'   wxo 1424
            ​*1.2.15  Truth tables: Operations on true and false constants   truantru 1450
            ​*1.2.16  Stoic logic indemonstrables (Chrysippus of Soli)   mptnan 1472
            1.2.17  Logical implication (continued)   syl6an 1483
      ​1.3  Predicate calculus mostly without distinct variables
            ​*1.3.1  Universal quantifier (continued)   ax-5 1500
            ​*1.3.2  Equality predicate (continued)   weq 1556
            1.3.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1579
            1.3.4  Introduce Axiom of Existence   ax-i9 1583
            1.3.5  Additional intuitionistic axioms   ax-ial 1587
            1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1589
            1.3.7  The existential quantifier   19.8a 1643
            1.3.8  Equality theorems without distinct variables   a9e 1748
            1.3.9  Axioms ax-10 and ax-11   ax10o 1767
            1.3.10  Substitution (without distinct variables)   wsb 1815
            1.3.11  Theorems using axiom ax-11   equs5a 1847
      ​1.4  Predicate calculus with distinct variables
            1.4.1  Derive the axiom of distinct variables ax-16   spimv 1864
            1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1875
            1.4.3  More theorems related to ax-11 and substitution   albidv 1877
            1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1911
            1.4.5  More substitution theorems   hbs1 1998
            1.4.6  Existential uniqueness   weu 2086
            ​*1.4.7  Aristotelian logic: Assertic syllogisms   barbara 2185
      ​​*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 2220
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2224
                  2.1.2.1  Elementary properties of class abstractions   eqabdv 2369
            2.1.3  Class form not-free predicate   wnfc 2379
            2.1.4  Negated equality and membership   wne 2420
                  2.1.4.1  Negated equality   wne 2420
                  2.1.4.2  Negated membership   wnel 2515
            2.1.5  Restricted quantification   wral 2528
            2.1.6  The universal class   cvv 2821
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3034
            2.1.8  Russell's Paradox   ru 3050
            2.1.9  Proper substitution of classes for sets   wsbc 3051
            2.1.10  Proper substitution of classes for sets into classes   csb 3147
            2.1.11  Define basic set operations and relations   cdif 3217
            2.1.12  Subclasses and subsets   df-ss 3233
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3339
                  2.1.13.1  The difference of two classes   dfdif3 3339
                  2.1.13.2  The union of two classes   elun 3370
                  2.1.13.3  The intersection of two classes   elin 3412
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3462
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3497
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3513
            2.1.14  The empty set   c0 3520
            2.1.15  Conditional operator   cif 3638
            2.1.16  Power classes   cpw 3688
            2.1.17  Unordered and ordered pairs   csn 3709
            2.1.18  The union of a class   cuni 3935
            2.1.19  The intersection of a class   cint 3970
            2.1.20  Indexed union and intersection   ciun 4012
            2.1.21  Disjointness   wdisj 4106
            2.1.22  Binary relations   wbr 4130
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4191
            2.1.24  Transitive classes   wtr 4229
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4246
            2.2.2  Introduce the Axiom of Separation   ax-sep 4249
            2.2.3  Derive the Null Set Axiom   zfnuleu 4257
            2.2.4  Theorems requiring subset and intersection existence   nalset 4263
            2.2.5  Theorems requiring empty set existence   class2seteq 4300
            2.2.6  Collection principle   bnd 4309
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4311
            2.3.2  A notation for excluded middle   wem 4331
            2.3.3  Axiom of Pairing   ax-pr 4346
            2.3.4  Ordered pair theorem   opm 4374
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4398
            2.3.6  Power class of union and intersection   pwin 4427
            2.3.7  Epsilon and identity relations   cep 4432
            ​*2.3.8  Partial and total orderings   wpo 4439
            2.3.9  Founded and set-like relations   wfrfor 4472
            2.3.10  Ordinals   word 4507
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4578
            2.4.2  Ordinals (continued)   ordon 4633
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4679
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4684
            2.5.3  Transfinite induction   tfi 4729
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4735
            2.6.2  The natural numbers   com 4737
            2.6.3  Peano's postulates   peano1 4741
            2.6.4  Finite induction (for finite ordinals)   find 4746
            2.6.5  The Natural Numbers (continued)   nn0suc 4751
            2.6.6  Relations   cxp 4772
            2.6.7  Definite description binder (inverted iota)   cio 5335
            2.6.8  Functions   wfun 5371
            2.6.9  Cantor's Theorem   canth 6036
            2.6.10  Restricted iota (description binder)   crio 6037
            2.6.11  Operations   co 6085
            2.6.12  Maps-to notation   elmpocl 6284
            2.6.13  Function operation   cof 6300
            2.6.14  Functions (continued)   resfunexgALT 6337
            2.6.15  First and second members of an ordered pair   c1st 6372
            ​*2.6.16  The support of functions   csupp 6475
            ​*2.6.17  Special maps-to operations   opeliunxp2f 6509
            2.6.18  Function transposition   ctpos 6515
            2.6.19  Undefined values   pwuninel2 6553
            2.6.20  Functions on ordinals; strictly monotone ordinal functions   iunon 6555
            2.6.21  "Strong" transfinite recursion   crecs 6575
            2.6.22  Recursive definition generator   crdg 6640
            2.6.23  Finite recursion   cfrec 6661
            2.6.24  Ordinal arithmetic   c1o 6680
            2.6.25  Natural number arithmetic   nna0 6747
            2.6.26  Equivalence relations and classes   wer 6804
            2.6.27  The mapping operation   cmap 6922
            2.6.28  Infinite Cartesian products   cixp 6980
            2.6.29  Equinumerosity   cen 7020
            2.6.30  Equinumerosity (cont.)   xpf1o 7144
            2.6.31  Pigeonhole Principle   phplem1 7153
            2.6.32  Finite sets   fict 7170
            2.6.33  Schroeder-Bernstein Theorem   sbthlem1 7274
            2.6.34  Finitely supported functions   cfsupp 7285
            2.6.35  Finite intersections   cfi 7302
            2.6.36  The sizes of sets   2omap 7318
            2.6.37  Supremum and infimum   csup 7322
            2.6.38  Ordinal isomorphism   ordiso2 7375
            2.6.39  Disjoint union   cdju 7377
                  2.6.39.1  Disjoint union   cdju 7377
                  ​*2.6.39.2  Left and right injections of a disjoint union   cinl 7385
                  2.6.39.3  Universal property of the disjoint union   djuss 7410
                  2.6.39.4  Dominance and equinumerosity properties of disjoint union   djudom 7433
                  2.6.39.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7442
                  2.6.39.6  Countable sets   0ct 7447
            ​*2.6.40  The one-point compactification of the natural numbers   xnninf 7459
            2.6.41  Omniscient sets   comni 7474
            2.6.42  Markov's principle   cmarkov 7491
            2.6.43  Weakly omniscient sets   cwomni 7503
            2.6.44  Cardinal numbers   ccrd 7522
            2.6.45  Axiom of Choice equivalents   wac 7561
            2.6.46  Cardinal number arithmetic   endjudisj 7566
            2.6.47  Ordinal trichotomy   exmidontriimlem1 7577
            2.6.48  Excluded middle and the power set of a singleton   iftrueb01 7582
            2.6.49  Apartness relations   wap 7607
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7628
​​*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 7639
            4.1.2  Final derivation of real and complex number postulates   axcnex 8226
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 8270
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 8303
            4.2.2  Infinity and the extended real number system   cpnf 8357
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8392
            4.2.4  Ordering on reals   lttr 8399
            4.2.5  Initial properties of the complex numbers   mul12 8455
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8484
            4.3.2  Subtraction   cmin 8497
            4.3.3  Multiplication   kcnktkm1cn 8710
            4.3.4  Ordering on reals (cont.)   ltadd2 8747
            4.3.5  Real Apartness   creap 8902
            4.3.6  Complex Apartness   cap 8909
            4.3.7  Reciprocals   recextlem1 8979
            4.3.8  Division   cdiv 9002
            4.3.9  Ordering on reals (cont.)   ltp1 9174
            4.3.10  Suprema   lbreu 9275
            4.3.11  Imaginary and complex number properties   crap0 9288
            4.3.12  Function operation analogue theorems   ofnegsub 9292
            ​*4.3.13  Indicator Functions   cind 9293
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 9304
            4.4.2  Principle of mathematical induction   nnind 9320
            ​*4.4.3  Decimal representation of numbers   c2 9355
            ​*4.4.4  Some properties of specific numbers   neg1cn 9409
            4.4.5  Simple number properties   halfcl 9531
            4.4.6  The Archimedean property   arch 9560
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9563
            ​*4.4.8  Extended nonnegative integers   cxnn0 9630
            4.4.9  Integers (as a subset of complex numbers)   cz 9644
            4.4.10  Decimal arithmetic   cdc 9777
            4.4.11  Upper sets of integers   cuz 9921
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 10019
            4.4.13  Complex numbers as pairs of reals   cnref1o 10051
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 10054
            4.5.2  Infinity and the extended real number system (cont.)   cxne 10171
            4.5.3  Real number intervals   cioo 10290
            4.5.4  Finite intervals of integers   cfz 10411
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10519
            4.5.6  Half-open integer ranges   cfzo 10549
            4.5.7  Rational numbers (cont.)   qtri3or 10675
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10703
            4.6.2  The modulo (remainder) operation   cmo 10759
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10836
            4.6.4  Strong induction over upper sets of integers   uzsinds 10881
            4.6.5  The infinite sequence builder "seq"   cseq 10884
            4.6.6  Integer powers   cexp 10975
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 11157
            4.6.8  Factorial function   cfa 11163
            4.6.9  The binomial coefficient operation   cbc 11185
            4.6.10  The ` # ` (set size) function   chash 11214
                  4.6.10.1  Proper unordered pairs and triples (sets of size 2 and 3)   hash2en 11295
                  4.6.10.2  Functions with a domain containing at least two different elements   fundm2domnop0 11300
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 11304
            4.7.2  Last symbol of a word   clsw 11349
            4.7.3  Concatenations of words   cconcat 11358
            4.7.4  Singleton words   cs1 11383
            4.7.5  Concatenations with singleton words   ccatws1cl 11400
            4.7.6  Subwords/substrings   csubstr 11417
            4.7.7  Prefixes of a word   cpfx 11444
            4.7.8  Subwords of subwords   swrdswrdlem 11476
            4.7.9  Subwords and concatenations   pfxcctswrd 11482
            4.7.10  Subwords of concatenations   swrdccatfn 11496
            4.7.11  Longer string literals   cs2 11521
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 11579
            4.8.2  Real and imaginary parts; conjugate   ccj 11604
            4.8.3  Sequence convergence   caucvgrelemrec 11745
            4.8.4  Square root; absolute value   csqrt 11762
            4.8.5  The maximum of two real numbers   maxcom 11969
            4.8.6  The minimum of two real numbers   mincom 11995
            4.8.7  The maximum of two extended reals   xrmaxleim 12010
            4.8.8  The minimum of two extended reals   xrnegiso 12028
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 12044
            4.9.2  Finite and infinite sums   csu 12119
            4.9.3  The binomial theorem   binomlem 12250
            4.9.4  Infinite sums (cont.)   isumshft 12257
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 12264
            4.9.6  Arithmetic series   arisum 12265
            4.9.7  Geometric series   expcnvap0 12269
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 12290
            4.9.9  Mertens' theorem   mertenslemub 12301
            4.9.10  Finite and infinite products   prodf 12305
                  4.9.10.1  Product sequences   prodf 12305
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 12315
                  4.9.10.3  Complex products   cprod 12317
                  4.9.10.4  Finite products   fprodseq 12350
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 12409
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 12542
            4.10.2  _e is irrational   eirraplem 12544
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 12554
            ​*5.1.2  Even and odd numbers   evenelz 12634
            5.1.3  The division algorithm   divalglemnn 12685
            5.1.4  Bit sequences   cbits 12707
            5.1.5  The greatest common divisor operator   cgcd 12730
            5.1.6  Bézout's identity   bezoutlemnewy 12773
            5.1.7  Decidable sets of integers   nnmindc 12811
            5.1.8  Algorithms   nn0seqcvgd 12819
            5.1.9  Euclid's Algorithm   eucalgval2 12831
            ​*5.1.10  The least common multiple   clcm 12838
            ​*5.1.11  Coprimality and Euclid's lemma   coprmgcdb 12866
            5.1.12  Cancellability of congruences   congr 12878
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12885
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12922
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12939
            5.2.4  Properties of the canonical representation of a rational   cnumer 12959
            5.2.5  Euler's theorem   codz 12986
            5.2.6  Arithmetic modulo a prime number   modprm1div 13026
            5.2.7  Pythagorean Triples   coprimeprodsq 13036
            5.2.8  The prime count function   cpc 13063
            5.2.9  Pocklington's theorem   prmpwdvds 13134
            5.2.10  Infinite primes theorem   infpnlem1 13138
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 13142
            5.2.12  Lagrange's four-square theorem   cgz 13148
            5.2.13  Decimal arithmetic (cont.)   dec2dvds 13190
            5.2.14  Bertrand's Ballot Problem   ballotfilemofi 13219
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 13283
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 13348
            6.1.2  Slot definitions   cplusg 13431
            6.1.3  Various definitions used by the structure product   crest 13593
            6.1.4  Definition of the structure quotient   cimas 13622
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 13673
            ​*7.1.2  Identity elements   mgmidmo 13692
            7.1.3  Iterated sums in a magma   fngzsum 13708
            ​*7.1.4  Semigroups   csgrp 13716
            ​*7.1.5  Definition and basic properties of monoids   cmnd 13729
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13764
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13800
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13805
            ​*7.2.2  Group multiple operation   cmg 13922
            7.2.3  Subgroups and Quotient groups   csubg 13970
            7.2.4  Elementary theory of group homomorphisms   cghm 14043
            7.2.5  Abelian groups   ccmn 14087
                  7.2.5.1  Definition and basic properties   ccmn 14087
                  7.2.5.2  Group sum operation   gzsumreidx 14141
            7.2.6  Finite group sum over unordered finite set   cgsu 14150
            7.2.7  Structure product   cprds 14169
            7.2.8  Binary product on structures   cxps 14199
            7.2.9  Structure power   cpws 14202
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 14217
            ​*7.3.2  Non-unital rings ("rngs")   crng 14231
            ​*7.3.3  Ring unity (multiplicative identity)   cur 14262
            7.3.4  Semirings   csrg 14267
            7.3.5  Definition and basic properties of unital rings   crg 14300
            7.3.6  Opposite ring   coppr 14372
            7.3.7  Divisibility   cdsr 14392
            7.3.8  Ring homomorphisms   crh 14457
            7.3.9  Nonzero rings and zero rings   cnzr 14486
            7.3.10  Local rings   clring 14497
            7.3.11  Subrings   csubrng 14505
                  7.3.11.1  Subrings of non-unital rings   csubrng 14505
                  7.3.11.2  Subrings of unital rings   csubrg 14525
            7.3.12  Left regular elements and domains   crlreg 14563
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 14589
            7.4.2  Definition and basic properties   cdr 14602
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 14623
            7.5.2  Subspaces and spans in a left module   clss 14689
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 14770
            7.6.2  Ideals and spans   clidl 14804
            7.6.3  Two-sided ideals and quotient rings   c2idl 14836
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14871
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14872
            ​*7.7.2  Ring of integers   czring 14925
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14946
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Associative algebras
            8.1.1  Definition and basic properties   casa 14996
      ​8.2  Abstract multivariate polynomials
            8.2.1  Definition and basic properties   cmps 15045
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 15098
                  9.1.1.1  Topologies   ctop 15098
                  9.1.1.2  Topologies on sets   ctopon 15111
                  9.1.1.3  Topological spaces   ctps 15131
            9.1.2  Topological bases   ctb 15143
            9.1.3  Examples of topologies   distop 15186
            9.1.4  Closure and interior   ccld 15193
            9.1.5  Neighborhoods   cnei 15239
            9.1.6  Subspace topologies   restrcl 15268
            9.1.7  Limits and continuity in topological spaces   ccn 15286
            9.1.8  Product topologies   ctx 15353
            9.1.9  Continuous function-builders   cnmptid 15382
            9.1.10  Homeomorphisms   chmeo 15401
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 15423
            9.2.2  Basic metric space properties   cxms 15437
            9.2.3  Metric space balls   blfvalps 15486
            9.2.4  Open sets of a metric space   mopnrel 15542
            9.2.5  Continuity in metric spaces   metcnp3 15612
            9.2.6  Topology on the reals   qtopbasss 15622
            9.2.7  Topological definitions using the reals   ccncf 15671
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 15718
            10.1.2  Intermediate value theorem   ivthinclemlm 15735
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 15755
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 15755
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 15829
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 15869
            11.2.2  Properties of pi = 3.14159...   pilem1 15880
            11.2.3  The natural logarithm on complex numbers   clog 15957
            ​*11.2.4  Logarithms to an arbitrary base   clogb 16045
            11.2.5  Quartic binomial expansion   binom4 16081
            11.2.6  Logarithms (cont.)   log2tlbndlog2 16082
            11.2.7  The Birthday Problem   log2ublem1 16083
      ​11.3  Pell equations
            11.3.1  Pell equations 1: A nontrivial solution always exists   pellexlem1 16091
      ​11.4  Basic number theory
            11.4.1  Wilson's theorem   wilthlem1 16094
            11.4.2  Number-theoretical functions   csgm 16095
            11.4.3  Perfect Number Theorem   mersenne 16111
            ​*11.4.4  Quadratic residues and the Legendre symbol   clgs 16116
            ​*11.4.5  Gauss' Lemma   gausslemma2dlem0a 16168
            11.4.6  Quadratic reciprocity   lgseisenlem1 16189
            11.4.7  All primes 4n+1 are the sum of two squares   2sqlem1 16233
​PART 12  GRAPH THEORY
      ​12.1  Vertices and edges
            12.1.1  The edge function extractor for extensible structures   cedgf 16245
            12.1.2  Vertices and indexed edges   cvtx 16253
                  12.1.2.1  Definitions and basic properties   cvtx 16253
                  12.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 16262
                  12.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdm2domval 16270
                  12.1.2.4  Degenerated cases of representations of graphs   vtxval0 16294
            12.1.3  Edges as range of the edge function   cedg 16298
      ​12.2  Undirected graphs
            12.2.1  Undirected hypergraphs   cuhgr 16308
            12.2.2  Undirected pseudographs and multigraphs   cupgr 16332
            ​*12.2.3  Loop-free graphs   umgrislfupgrenlem 16371
            12.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 16375
            ​*12.2.5  Undirected simple graphs   cuspgr 16394
            12.2.6  Examples for graphs   usgr0e 16473
            12.2.7  Subgraphs   csubgr 16494
            12.2.8  Vertex degree   cvtxdg 16527
      ​12.3  Walks, paths and cycles
            12.3.1  Walks   cwlks 16558
            12.3.2  Trails   ctrls 16621
            12.3.3  Closed walks as words   cclwwlk 16632
                  12.3.3.1  Closed walks as words   cclwwlk 16632
                  12.3.3.2  Closed walks of a fixed length as words   cclwwlkn 16644
                  12.3.3.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 16667
      ​12.4  Eulerian paths and the Konigsberg Bridge problem
            ​*12.4.1  Eulerian paths   ceupth 16683
            ​*12.4.2  The Königsberg Bridge problem   konigsbergvtx 16723
​PART 13  GUIDES AND MISCELLANEA
      ​13.1  Guides (conventions, explanations, and examples)
            ​*13.1.1  Conventions   conventions 16735
            13.1.2  Definitional examples   ex-or 16736
​PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​14.1  Mathboxes for user contributions
            14.1.1  Mathbox guidelines   mathbox 16746
      ​14.2  Mathbox for Matthew House
      ​14.3  Mathbox for BJ
            14.3.1  Propositional calculus   bj-nnsn 16761
                  ​*14.3.1.1  Stable formulas   bj-trst 16767
                  14.3.1.2  Decidable formulas   bj-trdc 16780
            14.3.2  Predicate calculus   bj-ex 16790
            14.3.3  Set theorey miscellaneous   bj-el2oss1o 16802
            ​*14.3.4  Extensionality   bj-vtoclgft 16803
            ​*14.3.5  Decidability of classes   wdcin 16821
            14.3.6  Disjoint union   djucllem 16828
            14.3.7  Miscellaneous   funmptd 16831
            ​*14.3.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 16838
                  *14.3.8.1  Bounded formulas   wbd 16838
                  ​*14.3.8.2  Bounded classes   wbdc 16866
            ​*14.3.9  CZF: Bounded separation   ax-bdsep 16910
                  14.3.9.1  Delta_0-classical logic   ax-bj-d0cl 16950
                  14.3.9.2  Inductive classes and the class of natural number ordinals   wind 16952
                  ​*14.3.9.3  The first three Peano postulates   bj-peano2 16965
            ​*14.3.10  CZF: Infinity   ax-infvn 16967
                  *14.3.10.1  The set of natural number ordinals   ax-infvn 16967
                  ​*14.3.10.2  Peano's fifth postulate   bdpeano5 16969
                  ​*14.3.10.3  Bounded induction and Peano's fourth postulate   findset 16971
            ​*14.3.11  CZF: Set induction   setindft 16991
                  *14.3.11.1  Set induction   setindft 16991
                  ​*14.3.11.2  Full induction   bj-findis 17005
            ​*14.3.12  CZF: Strong collection   ax-strcoll 17008
            ​*14.3.13  CZF: Subset collection   ax-sscoll 17013
            14.3.14  Real numbers   ax-ddkcomp 17015
      ​14.4  Mathbox for Jim Kingdon
            14.4.1  Propositional and predicate logic   nnnotnotr 17016
            14.4.2  The sizes of sets   ss1oel2o 17017
            14.4.3  The power set of a singleton   pwtrufal 17027
            14.4.4  Weak excluded middle   wwem 17040
            14.4.5  Omniscience of NN+oo   0nninf 17047
            14.4.6  Schroeder-Bernstein Theorem   exmidsbthrlem 17067
            14.4.7  Real and complex numbers   qdencn 17072
            ​*14.4.8  Analytic omniscience principles   trilpolemclim 17085
            14.4.9  Supremum and infimum   supfz 17121
            14.4.10  Circle constant   taupi 17123
      ​14.5  Mathbox for Mykola Mostovenko
      ​14.6  Mathbox for David A. Wheeler
            14.6.1  Testable propositions   dftest 17125
            ​*14.6.2  Allsome quantifier   wals 17126
            ​*14.6.3  Allsome one quantifier   walseu 17160