TOC of Theorem List - Intuitionistic Logic Explorer

Table of Contents Summary

PART 1  INTUITIONISTIC FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Predicate calculus mostly without distinct variables
      1.4  Predicate calculus with distinct variables
      1.5  First-order logic with one non-logical binary predicate
PART 2  SET THEORY
      2.1  IZF Set Theory - start with the Axiom of Extensionality
      2.2  IZF Set Theory - add the Axioms of Collection and Separation
      2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
      2.4  IZF Set Theory - add the Axiom of Union
      2.5  IZF Set Theory - add the Axiom of Set Induction
      2.6  IZF Set Theory - add the Axiom of Infinity
PART 3  CHOICE PRINCIPLES
      3.1  Countable Choice and Dependent Choice
PART 4  REAL AND COMPLEX NUMBERS
      4.1  Construction and axiomatization of real and complex numbers
      4.2  Derive the basic properties from the field axioms
      4.3  Real and complex numbers - basic operations
      4.4  Integer sets
      4.5  Order sets
      4.6  Elementary integer functions
      4.7  Words over a set
      4.8  Elementary real and complex functions
      4.9  Elementary limits and convergence
      4.10  Elementary trigonometry
PART 5  ELEMENTARY NUMBER THEORY
      5.1  Elementary properties of divisibility
      5.2  Elementary prime number theory
      5.3  Cardinality of real and complex number subsets
PART 6  BASIC STRUCTURES
      6.1  Extensible structures
PART 7  BASIC ALGEBRAIC STRUCTURES
      7.1  Monoids
      7.2  Groups
      7.3  Rings
      7.4  Division rings and fields
      7.5  Left modules
      7.6  Subring algebras and ideals
      7.7  The complex numbers as an algebraic extensible structure
PART 8  BASIC LINEAR ALGEBRA
      8.1  Abstract multivariate polynomials
PART 9  BASIC TOPOLOGY
      9.1  Topology
      9.2  Metric spaces
PART 10  BASIC REAL AND COMPLEX ANALYSIS
      10.1  Continuity
      10.2  Derivatives
PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      11.1  Polynomials
      11.2  Basic trigonometry
      11.3  Basic number theory
PART 12  GRAPH THEORY
      12.1  Vertices and edges
      12.2  Undirected graphs
PART 13  GUIDES AND MISCELLANEA
      13.1  Guides (conventions, explanations, and examples)
PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​14.1  Mathboxes for user contributions
      14.2  Mathbox for BJ
      14.3  Mathbox for Jim Kingdon
      14.4  Mathbox for Mykola Mostovenko
      14.5  Mathbox for David A. Wheeler

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

​​*PART 1  INTUITIONISTIC FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            1.2.2  Propositional logic axioms for implication   ax-mp 5
            ​*1.2.3  Logical implication   mp2b 8
            1.2.4  Logical conjunction and logical equivalence   wa 104
            1.2.5  Logical negation (intuitionistic)   ax-in1 615
            1.2.6  Logical disjunction   wo 710
            1.2.7  Stable propositions   wstab 832
            1.2.8  Decidable propositions   wdc 836
            ​*1.2.9  Theorems of decidable propositions   const 854
            1.2.10  Miscellaneous theorems of propositional calculus   pm5.21nd 918
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 980
            1.2.12  True and false constants   wal 1371
                  ​*1.2.12.1  Universal quantifier for use by df-tru   wal 1371
                  ​*1.2.12.2  Equality predicate for use by df-tru   cv 1372
                  1.2.12.3  Define the true and false constants   wtru 1374
            1.2.13  Logical 'xor'   wxo 1395
            ​*1.2.14  Truth tables: Operations on true and false constants   truantru 1421
            ​*1.2.15  Stoic logic indemonstrables (Chrysippus of Soli)   mptnan 1443
            1.2.16  Logical implication (continued)   syl6an 1454
      ​1.3  Predicate calculus mostly without distinct variables
            ​*1.3.1  Universal quantifier (continued)   ax-5 1471
            ​*1.3.2  Equality predicate (continued)   weq 1527
            1.3.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1550
            1.3.4  Introduce Axiom of Existence   ax-i9 1554
            1.3.5  Additional intuitionistic axioms   ax-ial 1558
            1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1560
            1.3.7  The existential quantifier   19.8a 1614
            1.3.8  Equality theorems without distinct variables   a9e 1720
            1.3.9  Axioms ax-10 and ax-11   ax10o 1739
            1.3.10  Substitution (without distinct variables)   wsb 1786
            1.3.11  Theorems using axiom ax-11   equs5a 1818
      ​1.4  Predicate calculus with distinct variables
            1.4.1  Derive the axiom of distinct variables ax-16   spimv 1835
            1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1846
            1.4.3  More theorems related to ax-11 and substitution   albidv 1848
            1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1882
            1.4.5  More substitution theorems   hbs1 1967
            1.4.6  Existential uniqueness   weu 2055
            ​*1.4.7  Aristotelian logic: Assertic syllogisms   barbara 2153
      ​​*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 2188
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2192
                  2.1.2.1  Elementary properties of class abstractions   eqabdv 2335
            2.1.3  Class form not-free predicate   wnfc 2336
            2.1.4  Negated equality and membership   wne 2377
                  2.1.4.1  Negated equality   wne 2377
                  2.1.4.2  Negated membership   wnel 2472
            2.1.5  Restricted quantification   wral 2485
            2.1.6  The universal class   cvv 2773
            ​*2.1.7  Conditional equality (experimental)   wcdeq 2982
            2.1.8  Russell's Paradox   ru 2998
            2.1.9  Proper substitution of classes for sets   wsbc 2999
            2.1.10  Proper substitution of classes for sets into classes   csb 3094
            2.1.11  Define basic set operations and relations   cdif 3164
            2.1.12  Subclasses and subsets   df-ss 3180
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3284
                  2.1.13.1  The difference of two classes   dfdif3 3284
                  2.1.13.2  The union of two classes   elun 3315
                  2.1.13.3  The intersection of two classes   elin 3357
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3405
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3440
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3454
            2.1.14  The empty set   c0 3461
            2.1.15  Conditional operator   cif 3572
            2.1.16  Power classes   cpw 3617
            2.1.17  Unordered and ordered pairs   csn 3634
            2.1.18  The union of a class   cuni 3852
            2.1.19  The intersection of a class   cint 3887
            2.1.20  Indexed union and intersection   ciun 3929
            2.1.21  Disjointness   wdisj 4023
            2.1.22  Binary relations   wbr 4047
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4108
            2.1.24  Transitive classes   wtr 4146
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4163
            2.2.2  Introduce the Axiom of Separation   ax-sep 4166
            2.2.3  Derive the Null Set Axiom   zfnuleu 4172
            2.2.4  Theorems requiring subset and intersection existence   nalset 4178
            2.2.5  Theorems requiring empty set existence   class2seteq 4211
            2.2.6  Collection principle   bnd 4220
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4222
            2.3.2  A notation for excluded middle   wem 4242
            2.3.3  Axiom of Pairing   ax-pr 4257
            2.3.4  Ordered pair theorem   opm 4282
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4306
            2.3.6  Power class of union and intersection   pwin 4333
            2.3.7  Epsilon and identity relations   cep 4338
            ​*2.3.8  Partial and total orderings   wpo 4345
            2.3.9  Founded and set-like relations   wfrfor 4378
            2.3.10  Ordinals   word 4413
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4484
            2.4.2  Ordinals (continued)   ordon 4538
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4584
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4589
            2.5.3  Transfinite induction   tfi 4634
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4640
            2.6.2  The natural numbers   com 4642
            2.6.3  Peano's postulates   peano1 4646
            2.6.4  Finite induction (for finite ordinals)   find 4651
            2.6.5  The Natural Numbers (continued)   nn0suc 4656
            2.6.6  Relations   cxp 4677
            2.6.7  Definite description binder (inverted iota)   cio 5235
            2.6.8  Functions   wfun 5270
            2.6.9  Cantor's Theorem   canth 5904
            2.6.10  Restricted iota (description binder)   crio 5905
            2.6.11  Operations   co 5951
            2.6.12  Maps-to notation   elmpocl 6148
            2.6.13  Function operation   cof 6163
            2.6.14  Functions (continued)   resfunexgALT 6200
            2.6.15  First and second members of an ordered pair   c1st 6231
            ​*2.6.16  Special maps-to operations   opeliunxp2f 6331
            2.6.17  Function transposition   ctpos 6337
            2.6.18  Undefined values   pwuninel2 6375
            2.6.19  Functions on ordinals; strictly monotone ordinal functions   iunon 6377
            2.6.20  "Strong" transfinite recursion   crecs 6397
            2.6.21  Recursive definition generator   crdg 6462
            2.6.22  Finite recursion   cfrec 6483
            2.6.23  Ordinal arithmetic   c1o 6502
            2.6.24  Natural number arithmetic   nna0 6567
            2.6.25  Equivalence relations and classes   wer 6624
            2.6.26  The mapping operation   cmap 6742
            2.6.27  Infinite Cartesian products   cixp 6792
            2.6.28  Equinumerosity   cen 6832
            2.6.29  Equinumerosity (cont.)   xpf1o 6948
            2.6.30  Pigeonhole Principle   phplem1 6956
            2.6.31  Finite sets   fict 6972
            2.6.32  Schroeder-Bernstein Theorem   sbthlem1 7066
            2.6.33  Finite intersections   cfi 7077
            2.6.34  Supremum and infimum   csup 7091
            2.6.35  Ordinal isomorphism   ordiso2 7144
            2.6.36  Disjoint union   cdju 7146
                  2.6.36.1  Disjoint union   cdju 7146
                  ​*2.6.36.2  Left and right injections of a disjoint union   cinl 7154
                  2.6.36.3  Universal property of the disjoint union   djuss 7179
                  2.6.36.4  Dominance and equinumerosity properties of disjoint union   djudom 7202
                  2.6.36.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7211
                  2.6.36.6  Countable sets   0ct 7216
            ​*2.6.37  The one-point compactification of the natural numbers   xnninf 7228
            2.6.38  Omniscient sets   comni 7243
            2.6.39  Markov's principle   cmarkov 7260
            2.6.40  Weakly omniscient sets   cwomni 7272
            2.6.41  Cardinal numbers   ccrd 7291
            2.6.42  Axiom of Choice equivalents   wac 7324
            2.6.43  Cardinal number arithmetic   endjudisj 7329
            2.6.44  Ordinal trichotomy   exmidontriimlem1 7340
            2.6.45  Excluded middle and the power set of a singleton   pw1on 7345
            2.6.46  Apartness relations   wap 7366
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7381
​​*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 7392
            4.1.2  Final derivation of real and complex number postulates   axcnex 7979
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 8023
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 8056
            4.2.2  Infinity and the extended real number system   cpnf 8111
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8146
            4.2.4  Ordering on reals   lttr 8153
            4.2.5  Initial properties of the complex numbers   mul12 8208
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8237
            4.3.2  Subtraction   cmin 8250
            4.3.3  Multiplication   kcnktkm1cn 8462
            4.3.4  Ordering on reals (cont.)   ltadd2 8499
            4.3.5  Real Apartness   creap 8654
            4.3.6  Complex Apartness   cap 8661
            4.3.7  Reciprocals   recextlem1 8731
            4.3.8  Division   cdiv 8752
            4.3.9  Ordering on reals (cont.)   ltp1 8924
            4.3.10  Suprema   lbreu 9025
            4.3.11  Imaginary and complex number properties   crap0 9038
            4.3.12  Function operation analogue theorems   ofnegsub 9042
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 9043
            4.4.2  Principle of mathematical induction   nnind 9059
            ​*4.4.3  Decimal representation of numbers   c2 9094
            ​*4.4.4  Some properties of specific numbers   neg1cn 9148
            4.4.5  Simple number properties   halfcl 9270
            4.4.6  The Archimedean property   arch 9299
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9302
            ​*4.4.8  Extended nonnegative integers   cxnn0 9365
            4.4.9  Integers (as a subset of complex numbers)   cz 9379
            4.4.10  Decimal arithmetic   cdc 9511
            4.4.11  Upper sets of integers   cuz 9655
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9747
            4.4.13  Complex numbers as pairs of reals   cnref1o 9779
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9782
            4.5.2  Infinity and the extended real number system (cont.)   cxne 9898
            4.5.3  Real number intervals   cioo 10017
            4.5.4  Finite intervals of integers   cfz 10137
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10241
            4.5.6  Half-open integer ranges   cfzo 10271
            4.5.7  Rational numbers (cont.)   qtri3or 10390
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10418
            4.6.2  The modulo (remainder) operation   cmo 10474
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10551
            4.6.4  Strong induction over upper sets of integers   uzsinds 10596
            4.6.5  The infinite sequence builder "seq"   cseq 10599
            4.6.6  Integer powers   cexp 10690
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10871
            4.6.8  Factorial function   cfa 10877
            4.6.9  The binomial coefficient operation   cbc 10899
            4.6.10  The ` # ` (set size) function   chash 10927
                  4.6.10.1  Proper unordered pairs and triples (sets of size 2 and 3)   hash2en 10995
                  4.6.10.2  Functions with a domain containing at least two different elements   fundm2domnop0 10997
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 11001
            4.7.2  Last symbol of a word   clsw 11045
            4.7.3  Concatenations of words   cconcat 11054
            4.7.4  Singleton words   cs1 11077
            4.7.5  Concatenations with singleton words   ccatws1cl 11094
            4.7.6  Subwords/substrings   csubstr 11106
            4.7.7  Prefixes of a word   cpfx 11133
            4.7.8  Subwords of subwords   swrdswrdlem 11163
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 11169
            4.8.2  Real and imaginary parts; conjugate   ccj 11194
            4.8.3  Sequence convergence   caucvgrelemrec 11334
            4.8.4  Square root; absolute value   csqrt 11351
            4.8.5  The maximum of two real numbers   maxcom 11558
            4.8.6  The minimum of two real numbers   mincom 11584
            4.8.7  The maximum of two extended reals   xrmaxleim 11599
            4.8.8  The minimum of two extended reals   xrnegiso 11617
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 11633
            4.9.2  Finite and infinite sums   csu 11708
            4.9.3  The binomial theorem   binomlem 11838
            4.9.4  Infinite sums (cont.)   isumshft 11845
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 11852
            4.9.6  Arithmetic series   arisum 11853
            4.9.7  Geometric series   expcnvap0 11857
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 11878
            4.9.9  Mertens' theorem   mertenslemub 11889
            4.9.10  Finite and infinite products   prodf 11893
                  4.9.10.1  Product sequences   prodf 11893
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 11903
                  4.9.10.3  Complex products   cprod 11905
                  4.9.10.4  Finite products   fprodseq 11938
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 11997
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 12130
            4.10.2  _e is irrational   eirraplem 12132
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 12142
            ​*5.1.2  Even and odd numbers   evenelz 12222
            5.1.3  The division algorithm   divalglemnn 12273
            5.1.4  Bit sequences   cbits 12295
            5.1.5  The greatest common divisor operator   cgcd 12318
            5.1.6  Bézout's identity   bezoutlemnewy 12361
            5.1.7  Decidable sets of integers   nnmindc 12399
            5.1.8  Algorithms   nn0seqcvgd 12407
            5.1.9  Euclid's Algorithm   eucalgval2 12419
            ​*5.1.10  The least common multiple   clcm 12426
            ​*5.1.11  Coprimality and Euclid's lemma   coprmgcdb 12454
            5.1.12  Cancellability of congruences   congr 12466
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12473
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12510
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12527
            5.2.4  Properties of the canonical representation of a rational   cnumer 12547
            5.2.5  Euler's theorem   codz 12574
            5.2.6  Arithmetic modulo a prime number   modprm1div 12614
            5.2.7  Pythagorean Triples   coprimeprodsq 12624
            5.2.8  The prime count function   cpc 12651
            5.2.9  Pocklington's theorem   prmpwdvds 12722
            5.2.10  Infinite primes theorem   infpnlem1 12726
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 12730
            5.2.12  Lagrange's four-square theorem   cgz 12736
            5.2.13  Decimal arithmetic (cont.)   dec2dvds 12778
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 12807
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 12872
            6.1.2  Slot definitions   cplusg 12953
            6.1.3  Definition of the structure product   crest 13115
            6.1.4  Definition of the structure quotient   cimas 13175
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 13229
            ​*7.1.2  Identity elements   mgmidmo 13248
            ​*7.1.3  Iterated sums in a magma   fngsum 13264
            ​*7.1.4  Semigroups   csgrp 13277
            ​*7.1.5  Definition and basic properties of monoids   cmnd 13292
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13333
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13369
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13376
            ​*7.2.2  Group multiple operation   cmg 13499
            7.2.3  Subgroups and Quotient groups   csubg 13547
            7.2.4  Elementary theory of group homomorphisms   cghm 13620
            7.2.5  Abelian groups   ccmn 13664
                  7.2.5.1  Definition and basic properties   ccmn 13664
                  7.2.5.2  Group sum operation   gsumfzreidx 13717
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 13726
            ​*7.3.2  Non-unital rings ("rngs")   crng 13738
            ​*7.3.3  Ring unity (multiplicative identity)   cur 13765
            7.3.4  Semirings   csrg 13769
            7.3.5  Definition and basic properties of unital rings   crg 13802
            7.3.6  Opposite ring   coppr 13873
            7.3.7  Divisibility   cdsr 13892
            7.3.8  Ring homomorphisms   crh 13956
            7.3.9  Nonzero rings and zero rings   cnzr 13985
            7.3.10  Local rings   clring 13996
            7.3.11  Subrings   csubrng 14003
                  7.3.11.1  Subrings of non-unital rings   csubrng 14003
                  7.3.11.2  Subrings of unital rings   csubrg 14023
            7.3.12  Left regular elements and domains   crlreg 14061
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 14086
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 14093
            7.5.2  Subspaces and spans in a left module   clss 14158
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 14239
            7.6.2  Ideals and spans   clidl 14273
            7.6.3  Two-sided ideals and quotient rings   c2idl 14305
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14340
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14341
            ​*7.7.2  Ring of integers   czring 14396
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14417
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Abstract multivariate polynomials
            8.1.1  Definition and basic properties   cmps 14467
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 14513
                  9.1.1.1  Topologies   ctop 14513
                  9.1.1.2  Topologies on sets   ctopon 14526
                  9.1.1.3  Topological spaces   ctps 14546
            9.1.2  Topological bases   ctb 14558
            9.1.3  Examples of topologies   distop 14601
            9.1.4  Closure and interior   ccld 14608
            9.1.5  Neighborhoods   cnei 14654
            9.1.6  Subspace topologies   restrcl 14683
            9.1.7  Limits and continuity in topological spaces   ccn 14701
            9.1.8  Product topologies   ctx 14768
            9.1.9  Continuous function-builders   cnmptid 14797
            9.1.10  Homeomorphisms   chmeo 14816
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 14838
            9.2.2  Basic metric space properties   cxms 14852
            9.2.3  Metric space balls   blfvalps 14901
            9.2.4  Open sets of a metric space   mopnrel 14957
            9.2.5  Continuity in metric spaces   metcnp3 15027
            9.2.6  Topology on the reals   qtopbasss 15037
            9.2.7  Topological definitions using the reals   ccncf 15086
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 15133
            10.1.2  Intermediate value theorem   ivthinclemlm 15150
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 15170
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 15170
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 15244
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 15284
            11.2.2  Properties of pi = 3.14159...   pilem1 15295
            11.2.3  The natural logarithm on complex numbers   clog 15372
            ​*11.2.4  Logarithms to an arbitrary base   clogb 15459
            11.2.5  Quartic binomial expansion   binom4 15495
      ​11.3  Basic number theory
            11.3.1  Wilson's theorem   wilthlem1 15496
            11.3.2  Number-theoretical functions   csgm 15497
            11.3.3  Perfect Number Theorem   mersenne 15513
            ​*11.3.4  Quadratic residues and the Legendre symbol   clgs 15518
            ​*11.3.5  Gauss' Lemma   gausslemma2dlem0a 15570
            11.3.6  Quadratic reciprocity   lgseisenlem1 15591
            11.3.7  All primes 4n+1 are the sum of two squares   2sqlem1 15635
​PART 12  GRAPH THEORY
      ​12.1  Vertices and edges
            12.1.1  The edge function extractor for extensible structures   cedgf 15647
            12.1.2  Vertices and indexed edges   cvtx 15655
                  12.1.2.1  Definitions and basic properties   cvtx 15655
                  12.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 15664
                  12.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdm2domval 15672
                  12.1.2.4  Degenerated cases of representations of graphs   vtxval0 15694
            12.1.3  Edges as range of the edge function   cedg 15698
      ​12.2  Undirected graphs
            12.2.1  Undirected hypergraphs   cuhgr 15707
​PART 13  GUIDES AND MISCELLANEA
      ​13.1  Guides (conventions, explanations, and examples)
            ​*13.1.1  Conventions   conventions 15731
            13.1.2  Definitional examples   ex-or 15732
​PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​14.1  Mathboxes for user contributions
            14.1.1  Mathbox guidelines   mathbox 15742
      ​14.2  Mathbox for BJ
            14.2.1  Propositional calculus   bj-nnsn 15743
                  ​*14.2.1.1  Stable formulas   bj-trst 15749
                  14.2.1.2  Decidable formulas   bj-trdc 15762
            14.2.2  Predicate calculus   bj-ex 15772
            14.2.3  Set theorey miscellaneous   bj-el2oss1o 15784
            ​*14.2.4  Extensionality   bj-vtoclgft 15785
            ​*14.2.5  Decidability of classes   wdcin 15803
            14.2.6  Disjoint union   djucllem 15810
            14.2.7  Miscellaneous   funmptd 15813
            ​*14.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 15822
                  *14.2.8.1  Bounded formulas   wbd 15822
                  ​*14.2.8.2  Bounded classes   wbdc 15850
            ​*14.2.9  CZF: Bounded separation   ax-bdsep 15894
                  14.2.9.1  Delta_0-classical logic   ax-bj-d0cl 15934
                  14.2.9.2  Inductive classes and the class of natural number ordinals   wind 15936
                  ​*14.2.9.3  The first three Peano postulates   bj-peano2 15949
            ​*14.2.10  CZF: Infinity   ax-infvn 15951
                  *14.2.10.1  The set of natural number ordinals   ax-infvn 15951
                  ​*14.2.10.2  Peano's fifth postulate   bdpeano5 15953
                  ​*14.2.10.3  Bounded induction and Peano's fourth postulate   findset 15955
            ​*14.2.11  CZF: Set induction   setindft 15975
                  *14.2.11.1  Set induction   setindft 15975
                  ​*14.2.11.2  Full induction   bj-findis 15989
            ​*14.2.12  CZF: Strong collection   ax-strcoll 15992
            ​*14.2.13  CZF: Subset collection   ax-sscoll 15997
            14.2.14  Real numbers   ax-ddkcomp 15999
      ​14.3  Mathbox for Jim Kingdon
            14.3.1  Propositional and predicate logic   nnnotnotr 16000
            14.3.2  Natural numbers   1dom1el 16001
            14.3.3  The power set of a singleton   pwtrufal 16008
            14.3.4  Omniscience of NN+oo   0nninf 16015
            14.3.5  Schroeder-Bernstein Theorem   exmidsbthrlem 16035
            14.3.6  Real and complex numbers   qdencn 16040
            ​*14.3.7  Analytic omniscience principles   trilpolemclim 16049
            14.3.8  Supremum and infimum   supfz 16084
            14.3.9  Circle constant   taupi 16086
      ​14.4  Mathbox for Mykola Mostovenko
      ​14.5  Mathbox for David A. Wheeler
            14.5.1  Testable propositions   dftest 16088
            ​*14.5.2  Allsome quantifier   walsi 16089