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  GUIDES AND MISCELLANEA
      12.1  Guides (conventions, explanations, and examples)
PART 13  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​13.1  Mathboxes for user contributions
      13.2  Mathbox for BJ
      13.3  Mathbox for Jim Kingdon
      13.4  Mathbox for Mykola Mostovenko
      13.5  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 709
            1.2.7  Stable propositions   wstab 831
            1.2.8  Decidable propositions   wdc 835
            ​*1.2.9  Theorems of decidable propositions   const 853
            1.2.10  Miscellaneous theorems of propositional calculus   pm5.21nd 917
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 979
            1.2.12  True and false constants   wal 1362
                  ​*1.2.12.1  Universal quantifier for use by df-tru   wal 1362
                  ​*1.2.12.2  Equality predicate for use by df-tru   cv 1363
                  1.2.12.3  Define the true and false constants   wtru 1365
            1.2.13  Logical 'xor'   wxo 1386
            ​*1.2.14  Truth tables: Operations on true and false constants   truantru 1412
            ​*1.2.15  Stoic logic indemonstrables (Chrysippus of Soli)   mptnan 1434
            1.2.16  Logical implication (continued)   syl6an 1445
      ​1.3  Predicate calculus mostly without distinct variables
            ​*1.3.1  Universal quantifier (continued)   ax-5 1458
            ​*1.3.2  Equality predicate (continued)   weq 1514
            1.3.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1537
            1.3.4  Introduce Axiom of Existence   ax-i9 1541
            1.3.5  Additional intuitionistic axioms   ax-ial 1545
            1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1547
            1.3.7  The existential quantifier   19.8a 1601
            1.3.8  Equality theorems without distinct variables   a9e 1707
            1.3.9  Axioms ax-10 and ax-11   ax10o 1726
            1.3.10  Substitution (without distinct variables)   wsb 1773
            1.3.11  Theorems using axiom ax-11   equs5a 1805
      ​1.4  Predicate calculus with distinct variables
            1.4.1  Derive the axiom of distinct variables ax-16   spimv 1822
            1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1833
            1.4.3  More theorems related to ax-11 and substitution   albidv 1835
            1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1869
            1.4.5  More substitution theorems   hbs1 1954
            1.4.6  Existential uniqueness   weu 2042
            ​*1.4.7  Aristotelian logic: Assertic syllogisms   barbara 2140
      ​​*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 2175
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2179
                  2.1.2.1  Elementary properties of class abstractions   eqabdv 2322
            2.1.3  Class form not-free predicate   wnfc 2323
            2.1.4  Negated equality and membership   wne 2364
                  2.1.4.1  Negated equality   wne 2364
                  2.1.4.2  Negated membership   wnel 2459
            2.1.5  Restricted quantification   wral 2472
            2.1.6  The universal class   cvv 2760
            ​*2.1.7  Conditional equality (experimental)   wcdeq 2969
            2.1.8  Russell's Paradox   ru 2985
            2.1.9  Proper substitution of classes for sets   wsbc 2986
            2.1.10  Proper substitution of classes for sets into classes   csb 3081
            2.1.11  Define basic set operations and relations   cdif 3151
            2.1.12  Subclasses and subsets   df-ss 3167
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3270
                  2.1.13.1  The difference of two classes   dfdif3 3270
                  2.1.13.2  The union of two classes   elun 3301
                  2.1.13.3  The intersection of two classes   elin 3343
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3391
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3426
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3440
            2.1.14  The empty set   c0 3447
            2.1.15  Conditional operator   cif 3558
            2.1.16  Power classes   cpw 3602
            2.1.17  Unordered and ordered pairs   csn 3619
            2.1.18  The union of a class   cuni 3836
            2.1.19  The intersection of a class   cint 3871
            2.1.20  Indexed union and intersection   ciun 3913
            2.1.21  Disjointness   wdisj 4007
            2.1.22  Binary relations   wbr 4030
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4090
            2.1.24  Transitive classes   wtr 4128
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4145
            2.2.2  Introduce the Axiom of Separation   ax-sep 4148
            2.2.3  Derive the Null Set Axiom   zfnuleu 4154
            2.2.4  Theorems requiring subset and intersection existence   nalset 4160
            2.2.5  Theorems requiring empty set existence   class2seteq 4193
            2.2.6  Collection principle   bnd 4202
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4204
            2.3.2  A notation for excluded middle   wem 4224
            2.3.3  Axiom of Pairing   ax-pr 4239
            2.3.4  Ordered pair theorem   opm 4264
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4287
            2.3.6  Power class of union and intersection   pwin 4314
            2.3.7  Epsilon and identity relations   cep 4319
            ​*2.3.8  Partial and total orderings   wpo 4326
            2.3.9  Founded and set-like relations   wfrfor 4359
            2.3.10  Ordinals   word 4394
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4465
            2.4.2  Ordinals (continued)   ordon 4519
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4565
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4570
            2.5.3  Transfinite induction   tfi 4615
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4621
            2.6.2  The natural numbers   com 4623
            2.6.3  Peano's postulates   peano1 4627
            2.6.4  Finite induction (for finite ordinals)   find 4632
            2.6.5  The Natural Numbers (continued)   nn0suc 4637
            2.6.6  Relations   cxp 4658
            2.6.7  Definite description binder (inverted iota)   cio 5214
            2.6.8  Functions   wfun 5249
            2.6.9  Cantor's Theorem   canth 5872
            2.6.10  Restricted iota (description binder)   crio 5873
            2.6.11  Operations   co 5919
            2.6.12  Maps-to notation   elmpocl 6115
            2.6.13  Function operation   cof 6130
            2.6.14  Functions (continued)   resfunexgALT 6162
            2.6.15  First and second members of an ordered pair   c1st 6193
            ​*2.6.16  Special maps-to operations   opeliunxp2f 6293
            2.6.17  Function transposition   ctpos 6299
            2.6.18  Undefined values   pwuninel2 6337
            2.6.19  Functions on ordinals; strictly monotone ordinal functions   iunon 6339
            2.6.20  "Strong" transfinite recursion   crecs 6359
            2.6.21  Recursive definition generator   crdg 6424
            2.6.22  Finite recursion   cfrec 6445
            2.6.23  Ordinal arithmetic   c1o 6464
            2.6.24  Natural number arithmetic   nna0 6529
            2.6.25  Equivalence relations and classes   wer 6586
            2.6.26  The mapping operation   cmap 6704
            2.6.27  Infinite Cartesian products   cixp 6754
            2.6.28  Equinumerosity   cen 6794
            2.6.29  Equinumerosity (cont.)   xpf1o 6902
            2.6.30  Pigeonhole Principle   phplem1 6910
            2.6.31  Finite sets   fict 6926
            2.6.32  Schroeder-Bernstein Theorem   sbthlem1 7018
            2.6.33  Finite intersections   cfi 7029
            2.6.34  Supremum and infimum   csup 7043
            2.6.35  Ordinal isomorphism   ordiso2 7096
            2.6.36  Disjoint union   cdju 7098
                  2.6.36.1  Disjoint union   cdju 7098
                  ​*2.6.36.2  Left and right injections of a disjoint union   cinl 7106
                  2.6.36.3  Universal property of the disjoint union   djuss 7131
                  2.6.36.4  Dominance and equinumerosity properties of disjoint union   djudom 7154
                  2.6.36.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7163
                  2.6.36.6  Countable sets   0ct 7168
            ​*2.6.37  The one-point compactification of the natural numbers   xnninf 7180
            2.6.38  Omniscient sets   comni 7195
            2.6.39  Markov's principle   cmarkov 7212
            2.6.40  Weakly omniscient sets   cwomni 7224
            2.6.41  Cardinal numbers   ccrd 7241
            2.6.42  Axiom of Choice equivalents   wac 7267
            2.6.43  Cardinal number arithmetic   endjudisj 7272
            2.6.44  Ordinal trichotomy   exmidontriimlem1 7283
            2.6.45  Excluded middle and the power set of a singleton   pw1on 7288
            2.6.46  Apartness relations   wap 7309
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7324
​​*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 7334
            4.1.2  Final derivation of real and complex number postulates   axcnex 7921
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 7965
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 7998
            4.2.2  Infinity and the extended real number system   cpnf 8053
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8088
            4.2.4  Ordering on reals   lttr 8095
            4.2.5  Initial properties of the complex numbers   mul12 8150
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8179
            4.3.2  Subtraction   cmin 8192
            4.3.3  Multiplication   kcnktkm1cn 8404
            4.3.4  Ordering on reals (cont.)   ltadd2 8440
            4.3.5  Real Apartness   creap 8595
            4.3.6  Complex Apartness   cap 8602
            4.3.7  Reciprocals   recextlem1 8672
            4.3.8  Division   cdiv 8693
            4.3.9  Ordering on reals (cont.)   ltp1 8865
            4.3.10  Suprema   lbreu 8966
            4.3.11  Imaginary and complex number properties   crap0 8979
            4.3.12  Function operation analogue theorems   ofnegsub 8983
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 8984
            4.4.2  Principle of mathematical induction   nnind 9000
            ​*4.4.3  Decimal representation of numbers   c2 9035
            ​*4.4.4  Some properties of specific numbers   neg1cn 9089
            4.4.5  Simple number properties   halfcl 9211
            4.4.6  The Archimedean property   arch 9240
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9243
            ​*4.4.8  Extended nonnegative integers   cxnn0 9306
            4.4.9  Integers (as a subset of complex numbers)   cz 9320
            4.4.10  Decimal arithmetic   cdc 9451
            4.4.11  Upper sets of integers   cuz 9595
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9687
            4.4.13  Complex numbers as pairs of reals   cnref1o 9719
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9722
            4.5.2  Infinity and the extended real number system (cont.)   cxne 9838
            4.5.3  Real number intervals   cioo 9957
            4.5.4  Finite intervals of integers   cfz 10077
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10181
            4.5.6  Half-open integer ranges   cfzo 10211
            4.5.7  Rational numbers (cont.)   qtri3or 10313
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10340
            4.6.2  The modulo (remainder) operation   cmo 10396
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10473
            4.6.4  Strong induction over upper sets of integers   uzsinds 10518
            4.6.5  The infinite sequence builder "seq"   cseq 10521
            4.6.6  Integer powers   cexp 10612
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10793
            4.6.8  Factorial function   cfa 10799
            4.6.9  The binomial coefficient operation   cbc 10821
            4.6.10  The ` # ` (set size) function   chash 10849
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 10917
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 10961
            4.8.2  Real and imaginary parts; conjugate   ccj 10986
            4.8.3  Sequence convergence   caucvgrelemrec 11126
            4.8.4  Square root; absolute value   csqrt 11143
            4.8.5  The maximum of two real numbers   maxcom 11350
            4.8.6  The minimum of two real numbers   mincom 11375
            4.8.7  The maximum of two extended reals   xrmaxleim 11390
            4.8.8  The minimum of two extended reals   xrnegiso 11408
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 11424
            4.9.2  Finite and infinite sums   csu 11499
            4.9.3  The binomial theorem   binomlem 11629
            4.9.4  Infinite sums (cont.)   isumshft 11636
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 11643
            4.9.6  Arithmetic series   arisum 11644
            4.9.7  Geometric series   expcnvap0 11648
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 11669
            4.9.9  Mertens' theorem   mertenslemub 11680
            4.9.10  Finite and infinite products   prodf 11684
                  4.9.10.1  Product sequences   prodf 11684
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 11694
                  4.9.10.3  Complex products   cprod 11696
                  4.9.10.4  Finite products   fprodseq 11729
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 11788
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 11921
            4.10.2  _e is irrational   eirraplem 11923
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 11933
            ​*5.1.2  Even and odd numbers   evenelz 12011
            5.1.3  The division algorithm   divalglemnn 12062
            5.1.4  The greatest common divisor operator   cgcd 12082
            5.1.5  Bézout's identity   bezoutlemnewy 12136
            5.1.6  Decidable sets of integers   nnmindc 12174
            5.1.7  Algorithms   nn0seqcvgd 12182
            5.1.8  Euclid's Algorithm   eucalgval2 12194
            ​*5.1.9  The least common multiple   clcm 12201
            ​*5.1.10  Coprimality and Euclid's lemma   coprmgcdb 12229
            5.1.11  Cancellability of congruences   congr 12241
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12248
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12285
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12302
            5.2.4  Properties of the canonical representation of a rational   cnumer 12322
            5.2.5  Euler's theorem   codz 12349
            5.2.6  Arithmetic modulo a prime number   modprm1div 12388
            5.2.7  Pythagorean Triples   coprimeprodsq 12398
            5.2.8  The prime count function   cpc 12425
            5.2.9  Pocklington's theorem   prmpwdvds 12496
            5.2.10  Infinite primes theorem   infpnlem1 12500
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 12504
            5.2.12  Lagrange's four-square theorem   cgz 12510
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 12552
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 12617
            6.1.2  Slot definitions   cplusg 12698
            6.1.3  Definition of the structure product   crest 12853
            6.1.4  Definition of the structure quotient   cimas 12885
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 12939
            ​*7.1.2  Identity elements   mgmidmo 12958
            ​*7.1.3  Iterated sums in a magma   fngsum 12974
            ​*7.1.4  Semigroups   csgrp 12987
            ​*7.1.5  Definition and basic properties of monoids   cmnd 13000
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13032
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13068
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13075
            ​*7.2.2  Group multiple operation   cmg 13192
            7.2.3  Subgroups and Quotient groups   csubg 13240
            7.2.4  Elementary theory of group homomorphisms   cghm 13313
            7.2.5  Abelian groups   ccmn 13357
                  7.2.5.1  Definition and basic properties   ccmn 13357
                  7.2.5.2  Group sum operation   gsumfzreidx 13410
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 13419
            ​*7.3.2  Non-unital rings ("rngs")   crng 13431
            ​*7.3.3  Ring unity (multiplicative identity)   cur 13458
            7.3.4  Semirings   csrg 13462
            7.3.5  Definition and basic properties of unital rings   crg 13495
            7.3.6  Opposite ring   coppr 13566
            7.3.7  Divisibility   cdsr 13585
            7.3.8  Ring homomorphisms   crh 13649
            7.3.9  Nonzero rings and zero rings   cnzr 13678
            7.3.10  Local rings   clring 13689
            7.3.11  Subrings   csubrng 13696
                  7.3.11.1  Subrings of non-unital rings   csubrng 13696
                  7.3.11.2  Subrings of unital rings   csubrg 13716
            7.3.12  Left regular elements and domains   crlreg 13754
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 13779
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 13786
            7.5.2  Subspaces and spans in a left module   clss 13851
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 13932
            7.6.2  Ideals and spans   clidl 13966
            7.6.3  Two-sided ideals and quotient rings   c2idl 13998
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14033
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14034
            ​*7.7.2  Ring of integers   czring 14089
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14110
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Abstract multivariate polynomials
            8.1.1  Definition and basic properties   cmps 14160
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 14176
                  9.1.1.1  Topologies   ctop 14176
                  9.1.1.2  Topologies on sets   ctopon 14189
                  9.1.1.3  Topological spaces   ctps 14209
            9.1.2  Topological bases   ctb 14221
            9.1.3  Examples of topologies   distop 14264
            9.1.4  Closure and interior   ccld 14271
            9.1.5  Neighborhoods   cnei 14317
            9.1.6  Subspace topologies   restrcl 14346
            9.1.7  Limits and continuity in topological spaces   ccn 14364
            9.1.8  Product topologies   ctx 14431
            9.1.9  Continuous function-builders   cnmptid 14460
            9.1.10  Homeomorphisms   chmeo 14479
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 14501
            9.2.2  Basic metric space properties   cxms 14515
            9.2.3  Metric space balls   blfvalps 14564
            9.2.4  Open sets of a metric space   mopnrel 14620
            9.2.5  Continuity in metric spaces   metcnp3 14690
            9.2.6  Topology on the reals   qtopbasss 14700
            9.2.7  Topological definitions using the reals   ccncf 14749
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 14796
            10.1.2  Intermediate value theorem   ivthinclemlm 14813
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 14833
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 14833
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 14907
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 14944
            11.2.2  Properties of pi = 3.14159...   pilem1 14955
            11.2.3  The natural logarithm on complex numbers   clog 15032
            ​*11.2.4  Logarithms to an arbitrary base   clogb 15116
            11.2.5  Quartic binomial expansion   binom4 15152
      ​11.3  Basic number theory
            11.3.1  Wilson's theorem   wilthlem1 15153
            ​*11.3.2  Quadratic residues and the Legendre symbol   clgs 15154
            ​*11.3.3  Gauss' Lemma   gausslemma2dlem0a 15206
            11.3.4  Quadratic reciprocity   lgseisenlem1 15227
            11.3.5  All primes 4n+1 are the sum of two squares   2sqlem1 15271
​PART 12  GUIDES AND MISCELLANEA
      ​12.1  Guides (conventions, explanations, and examples)
            ​*12.1.1  Conventions   conventions 15283
            12.1.2  Definitional examples   ex-or 15284
​PART 13  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​13.1  Mathboxes for user contributions
            13.1.1  Mathbox guidelines   mathbox 15294
      ​13.2  Mathbox for BJ
            13.2.1  Propositional calculus   bj-nnsn 15295
                  ​*13.2.1.1  Stable formulas   bj-trst 15301
                  13.2.1.2  Decidable formulas   bj-trdc 15314
            13.2.2  Predicate calculus   bj-ex 15324
            13.2.3  Set theorey miscellaneous   bj-el2oss1o 15336
            ​*13.2.4  Extensionality   bj-vtoclgft 15337
            ​*13.2.5  Decidability of classes   wdcin 15355
            13.2.6  Disjoint union   djucllem 15362
            13.2.7  Miscellaneous   funmptd 15365
            ​*13.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 15374
                  *13.2.8.1  Bounded formulas   wbd 15374
                  ​*13.2.8.2  Bounded classes   wbdc 15402
            ​*13.2.9  CZF: Bounded separation   ax-bdsep 15446
                  13.2.9.1  Delta_0-classical logic   ax-bj-d0cl 15486
                  13.2.9.2  Inductive classes and the class of natural number ordinals   wind 15488
                  ​*13.2.9.3  The first three Peano postulates   bj-peano2 15501
            ​*13.2.10  CZF: Infinity   ax-infvn 15503
                  *13.2.10.1  The set of natural number ordinals   ax-infvn 15503
                  ​*13.2.10.2  Peano's fifth postulate   bdpeano5 15505
                  ​*13.2.10.3  Bounded induction and Peano's fourth postulate   findset 15507
            ​*13.2.11  CZF: Set induction   setindft 15527
                  *13.2.11.1  Set induction   setindft 15527
                  ​*13.2.11.2  Full induction   bj-findis 15541
            ​*13.2.12  CZF: Strong collection   ax-strcoll 15544
            ​*13.2.13  CZF: Subset collection   ax-sscoll 15549
            13.2.14  Real numbers   ax-ddkcomp 15551
      ​13.3  Mathbox for Jim Kingdon
            13.3.1  Propositional and predicate logic   nnnotnotr 15552
            13.3.2  Natural numbers   1dom1el 15553
            13.3.3  The power set of a singleton   pwtrufal 15558
            13.3.4  Omniscience of NN+oo   0nninf 15564
            13.3.5  Schroeder-Bernstein Theorem   exmidsbthrlem 15582
            13.3.6  Real and complex numbers   qdencn 15587
            ​*13.3.7  Analytic omniscience principles   trilpolemclim 15596
            13.3.8  Supremum and infimum   supfz 15631
            13.3.9  Circle constant   taupi 15633
      ​13.4  Mathbox for Mykola Mostovenko
      ​13.5  Mathbox for David A. Wheeler
            13.5.1  Testable propositions   dftest 15635
            ​*13.5.2  Allsome quantifier   walsi 15636