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  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 3708
            2.1.18  The union of a class   cuni 3933
            2.1.19  The intersection of a class   cint 3968
            2.1.20  Indexed union and intersection   ciun 4010
            2.1.21  Disjointness   wdisj 4104
            2.1.22  Binary relations   wbr 4128
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4189
            2.1.24  Transitive classes   wtr 4227
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4244
            2.2.2  Introduce the Axiom of Separation   ax-sep 4247
            2.2.3  Derive the Null Set Axiom   zfnuleu 4255
            2.2.4  Theorems requiring subset and intersection existence   nalset 4261
            2.2.5  Theorems requiring empty set existence   class2seteq 4298
            2.2.6  Collection principle   bnd 4307
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4309
            2.3.2  A notation for excluded middle   wem 4329
            2.3.3  Axiom of Pairing   ax-pr 4344
            2.3.4  Ordered pair theorem   opm 4372
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4396
            2.3.6  Power class of union and intersection   pwin 4425
            2.3.7  Epsilon and identity relations   cep 4430
            ​*2.3.8  Partial and total orderings   wpo 4437
            2.3.9  Founded and set-like relations   wfrfor 4470
            2.3.10  Ordinals   word 4505
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4576
            2.4.2  Ordinals (continued)   ordon 4631
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4677
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4682
            2.5.3  Transfinite induction   tfi 4727
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4733
            2.6.2  The natural numbers   com 4735
            2.6.3  Peano's postulates   peano1 4739
            2.6.4  Finite induction (for finite ordinals)   find 4744
            2.6.5  The Natural Numbers (continued)   nn0suc 4749
            2.6.6  Relations   cxp 4770
            2.6.7  Definite description binder (inverted iota)   cio 5333
            2.6.8  Functions   wfun 5369
            2.6.9  Cantor's Theorem   canth 6029
            2.6.10  Restricted iota (description binder)   crio 6030
            2.6.11  Operations   co 6078
            2.6.12  Maps-to notation   elmpocl 6277
            2.6.13  Function operation   cof 6293
            2.6.14  Functions (continued)   resfunexgALT 6330
            2.6.15  First and second members of an ordered pair   c1st 6365
            ​*2.6.16  The support of functions   csupp 6468
            ​*2.6.17  Special maps-to operations   opeliunxp2f 6502
            2.6.18  Function transposition   ctpos 6508
            2.6.19  Undefined values   pwuninel2 6546
            2.6.20  Functions on ordinals; strictly monotone ordinal functions   iunon 6548
            2.6.21  "Strong" transfinite recursion   crecs 6568
            2.6.22  Recursive definition generator   crdg 6633
            2.6.23  Finite recursion   cfrec 6654
            2.6.24  Ordinal arithmetic   c1o 6673
            2.6.25  Natural number arithmetic   nna0 6740
            2.6.26  Equivalence relations and classes   wer 6797
            2.6.27  The mapping operation   cmap 6915
            2.6.28  Infinite Cartesian products   cixp 6973
            2.6.29  Equinumerosity   cen 7013
            2.6.30  Equinumerosity (cont.)   xpf1o 7137
            2.6.31  Pigeonhole Principle   phplem1 7146
            2.6.32  Finite sets   fict 7163
            2.6.33  Schroeder-Bernstein Theorem   sbthlem1 7267
            2.6.34  Finitely supported functions   cfsupp 7278
            2.6.35  Finite intersections   cfi 7295
            2.6.36  The sizes of sets   2omap 7311
            2.6.37  Supremum and infimum   csup 7315
            2.6.38  Ordinal isomorphism   ordiso2 7368
            2.6.39  Disjoint union   cdju 7370
                  2.6.39.1  Disjoint union   cdju 7370
                  ​*2.6.39.2  Left and right injections of a disjoint union   cinl 7378
                  2.6.39.3  Universal property of the disjoint union   djuss 7403
                  2.6.39.4  Dominance and equinumerosity properties of disjoint union   djudom 7426
                  2.6.39.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7435
                  2.6.39.6  Countable sets   0ct 7440
            ​*2.6.40  The one-point compactification of the natural numbers   xnninf 7452
            2.6.41  Omniscient sets   comni 7467
            2.6.42  Markov's principle   cmarkov 7484
            2.6.43  Weakly omniscient sets   cwomni 7496
            2.6.44  Cardinal numbers   ccrd 7515
            2.6.45  Axiom of Choice equivalents   wac 7554
            2.6.46  Cardinal number arithmetic   endjudisj 7559
            2.6.47  Ordinal trichotomy   exmidontriimlem1 7570
            2.6.48  Excluded middle and the power set of a singleton   iftrueb01 7575
            2.6.49  Apartness relations   wap 7600
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7621
​​*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 7632
            4.1.2  Final derivation of real and complex number postulates   axcnex 8219
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 8263
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 8296
            4.2.2  Infinity and the extended real number system   cpnf 8350
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8385
            4.2.4  Ordering on reals   lttr 8392
            4.2.5  Initial properties of the complex numbers   mul12 8448
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8477
            4.3.2  Subtraction   cmin 8490
            4.3.3  Multiplication   kcnktkm1cn 8703
            4.3.4  Ordering on reals (cont.)   ltadd2 8740
            4.3.5  Real Apartness   creap 8895
            4.3.6  Complex Apartness   cap 8902
            4.3.7  Reciprocals   recextlem1 8972
            4.3.8  Division   cdiv 8995
            4.3.9  Ordering on reals (cont.)   ltp1 9167
            4.3.10  Suprema   lbreu 9268
            4.3.11  Imaginary and complex number properties   crap0 9281
            4.3.12  Function operation analogue theorems   ofnegsub 9285
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 9286
            4.4.2  Principle of mathematical induction   nnind 9302
            ​*4.4.3  Decimal representation of numbers   c2 9337
            ​*4.4.4  Some properties of specific numbers   neg1cn 9391
            4.4.5  Simple number properties   halfcl 9513
            4.4.6  The Archimedean property   arch 9542
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9545
            ​*4.4.8  Extended nonnegative integers   cxnn0 9612
            4.4.9  Integers (as a subset of complex numbers)   cz 9626
            4.4.10  Decimal arithmetic   cdc 9759
            4.4.11  Upper sets of integers   cuz 9903
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 10001
            4.4.13  Complex numbers as pairs of reals   cnref1o 10033
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 10036
            4.5.2  Infinity and the extended real number system (cont.)   cxne 10153
            4.5.3  Real number intervals   cioo 10272
            4.5.4  Finite intervals of integers   cfz 10393
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10500
            4.5.6  Half-open integer ranges   cfzo 10530
            4.5.7  Rational numbers (cont.)   qtri3or 10656
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10684
            4.6.2  The modulo (remainder) operation   cmo 10740
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10817
            4.6.4  Strong induction over upper sets of integers   uzsinds 10862
            4.6.5  The infinite sequence builder "seq"   cseq 10865
            4.6.6  Integer powers   cexp 10956
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 11138
            4.6.8  Factorial function   cfa 11144
            4.6.9  The binomial coefficient operation   cbc 11166
            4.6.10  The ` # ` (set size) function   chash 11195
                  4.6.10.1  Proper unordered pairs and triples (sets of size 2 and 3)   hash2en 11276
                  4.6.10.2  Functions with a domain containing at least two different elements   fundm2domnop0 11281
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 11285
            4.7.2  Last symbol of a word   clsw 11330
            4.7.3  Concatenations of words   cconcat 11339
            4.7.4  Singleton words   cs1 11364
            4.7.5  Concatenations with singleton words   ccatws1cl 11381
            4.7.6  Subwords/substrings   csubstr 11398
            4.7.7  Prefixes of a word   cpfx 11425
            4.7.8  Subwords of subwords   swrdswrdlem 11457
            4.7.9  Subwords and concatenations   pfxcctswrd 11463
            4.7.10  Subwords of concatenations   swrdccatfn 11477
            4.7.11  Longer string literals   cs2 11502
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 11560
            4.8.2  Real and imaginary parts; conjugate   ccj 11585
            4.8.3  Sequence convergence   caucvgrelemrec 11726
            4.8.4  Square root; absolute value   csqrt 11743
            4.8.5  The maximum of two real numbers   maxcom 11950
            4.8.6  The minimum of two real numbers   mincom 11976
            4.8.7  The maximum of two extended reals   xrmaxleim 11991
            4.8.8  The minimum of two extended reals   xrnegiso 12009
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 12025
            4.9.2  Finite and infinite sums   csu 12100
            4.9.3  The binomial theorem   binomlem 12231
            4.9.4  Infinite sums (cont.)   isumshft 12238
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 12245
            4.9.6  Arithmetic series   arisum 12246
            4.9.7  Geometric series   expcnvap0 12250
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 12271
            4.9.9  Mertens' theorem   mertenslemub 12282
            4.9.10  Finite and infinite products   prodf 12286
                  4.9.10.1  Product sequences   prodf 12286
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 12296
                  4.9.10.3  Complex products   cprod 12298
                  4.9.10.4  Finite products   fprodseq 12331
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 12390
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 12523
            4.10.2  _e is irrational   eirraplem 12525
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 12535
            ​*5.1.2  Even and odd numbers   evenelz 12615
            5.1.3  The division algorithm   divalglemnn 12666
            5.1.4  Bit sequences   cbits 12688
            5.1.5  The greatest common divisor operator   cgcd 12711
            5.1.6  Bézout's identity   bezoutlemnewy 12754
            5.1.7  Decidable sets of integers   nnmindc 12792
            5.1.8  Algorithms   nn0seqcvgd 12800
            5.1.9  Euclid's Algorithm   eucalgval2 12812
            ​*5.1.10  The least common multiple   clcm 12819
            ​*5.1.11  Coprimality and Euclid's lemma   coprmgcdb 12847
            5.1.12  Cancellability of congruences   congr 12859
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12866
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12903
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12920
            5.2.4  Properties of the canonical representation of a rational   cnumer 12940
            5.2.5  Euler's theorem   codz 12967
            5.2.6  Arithmetic modulo a prime number   modprm1div 13007
            5.2.7  Pythagorean Triples   coprimeprodsq 13017
            5.2.8  The prime count function   cpc 13044
            5.2.9  Pocklington's theorem   prmpwdvds 13115
            5.2.10  Infinite primes theorem   infpnlem1 13119
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 13123
            5.2.12  Lagrange's four-square theorem   cgz 13129
            5.2.13  Decimal arithmetic (cont.)   dec2dvds 13171
            5.2.14  Bertrand's Ballot Problem   ballotfilemofi 13200
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 13264
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 13329
            6.1.2  Slot definitions   cplusg 13411
            6.1.3  Various definitions used by the structure product   crest 13573
            6.1.4  Definition of the structure quotient   cimas 13602
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 13653
            ​*7.1.2  Identity elements   mgmidmo 13672
            7.1.3  Iterated sums in a magma   fngzsum 13688
            ​*7.1.4  Semigroups   csgrp 13696
            ​*7.1.5  Definition and basic properties of monoids   cmnd 13709
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13744
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13780
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13785
            ​*7.2.2  Group multiple operation   cmg 13902
            7.2.3  Subgroups and Quotient groups   csubg 13950
            7.2.4  Elementary theory of group homomorphisms   cghm 14023
            7.2.5  Abelian groups   ccmn 14067
                  7.2.5.1  Definition and basic properties   ccmn 14067
                  7.2.5.2  Group sum operation   gzsumreidx 14121
            7.2.6  Finite group sum over unordered finite set   cgsu 14130
            7.2.7  Structure product   cprds 14149
            7.2.8  Binary product on structures   cxps 14179
            7.2.9  Structure power   cpws 14182
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 14197
            ​*7.3.2  Non-unital rings ("rngs")   crng 14209
            ​*7.3.3  Ring unity (multiplicative identity)   cur 14240
            7.3.4  Semirings   csrg 14244
            7.3.5  Definition and basic properties of unital rings   crg 14277
            7.3.6  Opposite ring   coppr 14348
            7.3.7  Divisibility   cdsr 14368
            7.3.8  Ring homomorphisms   crh 14433
            7.3.9  Nonzero rings and zero rings   cnzr 14462
            7.3.10  Local rings   clring 14473
            7.3.11  Subrings   csubrng 14481
                  7.3.11.1  Subrings of non-unital rings   csubrng 14481
                  7.3.11.2  Subrings of unital rings   csubrg 14501
            7.3.12  Left regular elements and domains   crlreg 14539
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 14565
            7.4.2  Definition and basic properties   cdr 14578
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 14599
            7.5.2  Subspaces and spans in a left module   clss 14664
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 14745
            7.6.2  Ideals and spans   clidl 14779
            7.6.3  Two-sided ideals and quotient rings   c2idl 14811
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14846
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14847
            ​*7.7.2  Ring of integers   czring 14900
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14921
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Abstract multivariate polynomials
            8.1.1  Definition and basic properties   cmps 14971
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 15024
                  9.1.1.1  Topologies   ctop 15024
                  9.1.1.2  Topologies on sets   ctopon 15037
                  9.1.1.3  Topological spaces   ctps 15057
            9.1.2  Topological bases   ctb 15069
            9.1.3  Examples of topologies   distop 15112
            9.1.4  Closure and interior   ccld 15119
            9.1.5  Neighborhoods   cnei 15165
            9.1.6  Subspace topologies   restrcl 15194
            9.1.7  Limits and continuity in topological spaces   ccn 15212
            9.1.8  Product topologies   ctx 15279
            9.1.9  Continuous function-builders   cnmptid 15308
            9.1.10  Homeomorphisms   chmeo 15327
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 15349
            9.2.2  Basic metric space properties   cxms 15363
            9.2.3  Metric space balls   blfvalps 15412
            9.2.4  Open sets of a metric space   mopnrel 15468
            9.2.5  Continuity in metric spaces   metcnp3 15538
            9.2.6  Topology on the reals   qtopbasss 15548
            9.2.7  Topological definitions using the reals   ccncf 15597
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 15644
            10.1.2  Intermediate value theorem   ivthinclemlm 15661
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 15681
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 15681
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 15755
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 15795
            11.2.2  Properties of pi = 3.14159...   pilem1 15806
            11.2.3  The natural logarithm on complex numbers   clog 15883
            ​*11.2.4  Logarithms to an arbitrary base   clogb 15971
            11.2.5  Quartic binomial expansion   binom4 16007
      ​11.3  Pell equations
            11.3.1  Pell equations 1: A nontrivial solution always exists   pellexlem1 16008
      ​11.4  Basic number theory
            11.4.1  Wilson's theorem   wilthlem1 16011
            11.4.2  Number-theoretical functions   csgm 16012
            11.4.3  Perfect Number Theorem   mersenne 16028
            ​*11.4.4  Quadratic residues and the Legendre symbol   clgs 16033
            ​*11.4.5  Gauss' Lemma   gausslemma2dlem0a 16085
            11.4.6  Quadratic reciprocity   lgseisenlem1 16106
            11.4.7  All primes 4n+1 are the sum of two squares   2sqlem1 16150
​PART 12  GRAPH THEORY
      ​12.1  Vertices and edges
            12.1.1  The edge function extractor for extensible structures   cedgf 16162
            12.1.2  Vertices and indexed edges   cvtx 16170
                  12.1.2.1  Definitions and basic properties   cvtx 16170
                  12.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 16179
                  12.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdm2domval 16187
                  12.1.2.4  Degenerated cases of representations of graphs   vtxval0 16211
            12.1.3  Edges as range of the edge function   cedg 16215
      ​12.2  Undirected graphs
            12.2.1  Undirected hypergraphs   cuhgr 16225
            12.2.2  Undirected pseudographs and multigraphs   cupgr 16249
            ​*12.2.3  Loop-free graphs   umgrislfupgrenlem 16288
            12.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 16292
            ​*12.2.5  Undirected simple graphs   cuspgr 16311
            12.2.6  Examples for graphs   usgr0e 16390
            12.2.7  Subgraphs   csubgr 16411
            12.2.8  Vertex degree   cvtxdg 16444
      ​12.3  Walks, paths and cycles
            12.3.1  Walks   cwlks 16475
            12.3.2  Trails   ctrls 16538
            12.3.3  Closed walks as words   cclwwlk 16549
                  12.3.3.1  Closed walks as words   cclwwlk 16549
                  12.3.3.2  Closed walks of a fixed length as words   cclwwlkn 16561
                  12.3.3.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 16584
      ​12.4  Eulerian paths and the Konigsberg Bridge problem
            ​*12.4.1  Eulerian paths   ceupth 16600
            ​*12.4.2  The Königsberg Bridge problem   konigsbergvtx 16640
​PART 13  GUIDES AND MISCELLANEA
      ​13.1  Guides (conventions, explanations, and examples)
            ​*13.1.1  Conventions   conventions 16652
            13.1.2  Definitional examples   ex-or 16653
​PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​14.1  Mathboxes for user contributions
            14.1.1  Mathbox guidelines   mathbox 16663
      ​14.2  Mathbox for Matthew House
      ​14.3  Mathbox for BJ
            14.3.1  Propositional calculus   bj-nnsn 16678
                  ​*14.3.1.1  Stable formulas   bj-trst 16684
                  14.3.1.2  Decidable formulas   bj-trdc 16697
            14.3.2  Predicate calculus   bj-ex 16707
            14.3.3  Set theorey miscellaneous   bj-el2oss1o 16719
            ​*14.3.4  Extensionality   bj-vtoclgft 16720
            ​*14.3.5  Decidability of classes   wdcin 16738
            14.3.6  Disjoint union   djucllem 16745
            14.3.7  Miscellaneous   funmptd 16748
            ​*14.3.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 16755
                  *14.3.8.1  Bounded formulas   wbd 16755
                  ​*14.3.8.2  Bounded classes   wbdc 16783
            ​*14.3.9  CZF: Bounded separation   ax-bdsep 16827
                  14.3.9.1  Delta_0-classical logic   ax-bj-d0cl 16867
                  14.3.9.2  Inductive classes and the class of natural number ordinals   wind 16869
                  ​*14.3.9.3  The first three Peano postulates   bj-peano2 16882
            ​*14.3.10  CZF: Infinity   ax-infvn 16884
                  *14.3.10.1  The set of natural number ordinals   ax-infvn 16884
                  ​*14.3.10.2  Peano's fifth postulate   bdpeano5 16886
                  ​*14.3.10.3  Bounded induction and Peano's fourth postulate   findset 16888
            ​*14.3.11  CZF: Set induction   setindft 16908
                  *14.3.11.1  Set induction   setindft 16908
                  ​*14.3.11.2  Full induction   bj-findis 16922
            ​*14.3.12  CZF: Strong collection   ax-strcoll 16925
            ​*14.3.13  CZF: Subset collection   ax-sscoll 16930
            14.3.14  Real numbers   ax-ddkcomp 16932
      ​14.4  Mathbox for Jim Kingdon
            14.4.1  Propositional and predicate logic   nnnotnotr 16933
            14.4.2  The sizes of sets   ss1oel2o 16934
            14.4.3  The power set of a singleton   pwtrufal 16944
            14.4.4  Omniscience of NN+oo   0nninf 16955
            14.4.5  Schroeder-Bernstein Theorem   exmidsbthrlem 16975
            14.4.6  Real and complex numbers   qdencn 16980
            ​*14.4.7  Analytic omniscience principles   trilpolemclim 16993
            14.4.8  Supremum and infimum   supfz 17029
            14.4.9  Circle constant   taupi 17031
      ​14.5  Mathbox for Mykola Mostovenko
      ​14.6  Mathbox for David A. Wheeler
            14.6.1  Testable propositions   dftest 17033
            ​*14.6.2  Allsome quantifier   wals 17034