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 1461
            ​*1.3.2  Equality predicate (continued)   weq 1517
            1.3.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1540
            1.3.4  Introduce Axiom of Existence   ax-i9 1544
            1.3.5  Additional intuitionistic axioms   ax-ial 1548
            1.3.6  Predicate calculus including ax-4, without distinct variables   spi 1550
            1.3.7  The existential quantifier   19.8a 1604
            1.3.8  Equality theorems without distinct variables   a9e 1710
            1.3.9  Axioms ax-10 and ax-11   ax10o 1729
            1.3.10  Substitution (without distinct variables)   wsb 1776
            1.3.11  Theorems using axiom ax-11   equs5a 1808
      ​1.4  Predicate calculus with distinct variables
            1.4.1  Derive the axiom of distinct variables ax-16   spimv 1825
            1.4.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1836
            1.4.3  More theorems related to ax-11 and substitution   albidv 1838
            1.4.4  Predicate calculus with distinct variables (cont.)   ax16i 1872
            1.4.5  More substitution theorems   hbs1 1957
            1.4.6  Existential uniqueness   weu 2045
            ​*1.4.7  Aristotelian logic: Assertic syllogisms   barbara 2143
      ​​*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 2178
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2182
                  2.1.2.1  Elementary properties of class abstractions   eqabdv 2325
            2.1.3  Class form not-free predicate   wnfc 2326
            2.1.4  Negated equality and membership   wne 2367
                  2.1.4.1  Negated equality   wne 2367
                  2.1.4.2  Negated membership   wnel 2462
            2.1.5  Restricted quantification   wral 2475
            2.1.6  The universal class   cvv 2763
            ​*2.1.7  Conditional equality (experimental)   wcdeq 2972
            2.1.8  Russell's Paradox   ru 2988
            2.1.9  Proper substitution of classes for sets   wsbc 2989
            2.1.10  Proper substitution of classes for sets into classes   csb 3084
            2.1.11  Define basic set operations and relations   cdif 3154
            2.1.12  Subclasses and subsets   df-ss 3170
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3274
                  2.1.13.1  The difference of two classes   dfdif3 3274
                  2.1.13.2  The union of two classes   elun 3305
                  2.1.13.3  The intersection of two classes   elin 3347
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3395
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3430
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3444
            2.1.14  The empty set   c0 3451
            2.1.15  Conditional operator   cif 3562
            2.1.16  Power classes   cpw 3606
            2.1.17  Unordered and ordered pairs   csn 3623
            2.1.18  The union of a class   cuni 3840
            2.1.19  The intersection of a class   cint 3875
            2.1.20  Indexed union and intersection   ciun 3917
            2.1.21  Disjointness   wdisj 4011
            2.1.22  Binary relations   wbr 4034
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4094
            2.1.24  Transitive classes   wtr 4132
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4149
            2.2.2  Introduce the Axiom of Separation   ax-sep 4152
            2.2.3  Derive the Null Set Axiom   zfnuleu 4158
            2.2.4  Theorems requiring subset and intersection existence   nalset 4164
            2.2.5  Theorems requiring empty set existence   class2seteq 4197
            2.2.6  Collection principle   bnd 4206
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4208
            2.3.2  A notation for excluded middle   wem 4228
            2.3.3  Axiom of Pairing   ax-pr 4243
            2.3.4  Ordered pair theorem   opm 4268
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4291
            2.3.6  Power class of union and intersection   pwin 4318
            2.3.7  Epsilon and identity relations   cep 4323
            ​*2.3.8  Partial and total orderings   wpo 4330
            2.3.9  Founded and set-like relations   wfrfor 4363
            2.3.10  Ordinals   word 4398
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4469
            2.4.2  Ordinals (continued)   ordon 4523
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4569
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4574
            2.5.3  Transfinite induction   tfi 4619
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4625
            2.6.2  The natural numbers   com 4627
            2.6.3  Peano's postulates   peano1 4631
            2.6.4  Finite induction (for finite ordinals)   find 4636
            2.6.5  The Natural Numbers (continued)   nn0suc 4641
            2.6.6  Relations   cxp 4662
            2.6.7  Definite description binder (inverted iota)   cio 5218
            2.6.8  Functions   wfun 5253
            2.6.9  Cantor's Theorem   canth 5878
            2.6.10  Restricted iota (description binder)   crio 5879
            2.6.11  Operations   co 5925
            2.6.12  Maps-to notation   elmpocl 6122
            2.6.13  Function operation   cof 6137
            2.6.14  Functions (continued)   resfunexgALT 6174
            2.6.15  First and second members of an ordered pair   c1st 6205
            ​*2.6.16  Special maps-to operations   opeliunxp2f 6305
            2.6.17  Function transposition   ctpos 6311
            2.6.18  Undefined values   pwuninel2 6349
            2.6.19  Functions on ordinals; strictly monotone ordinal functions   iunon 6351
            2.6.20  "Strong" transfinite recursion   crecs 6371
            2.6.21  Recursive definition generator   crdg 6436
            2.6.22  Finite recursion   cfrec 6457
            2.6.23  Ordinal arithmetic   c1o 6476
            2.6.24  Natural number arithmetic   nna0 6541
            2.6.25  Equivalence relations and classes   wer 6598
            2.6.26  The mapping operation   cmap 6716
            2.6.27  Infinite Cartesian products   cixp 6766
            2.6.28  Equinumerosity   cen 6806
            2.6.29  Equinumerosity (cont.)   xpf1o 6914
            2.6.30  Pigeonhole Principle   phplem1 6922
            2.6.31  Finite sets   fict 6938
            2.6.32  Schroeder-Bernstein Theorem   sbthlem1 7032
            2.6.33  Finite intersections   cfi 7043
            2.6.34  Supremum and infimum   csup 7057
            2.6.35  Ordinal isomorphism   ordiso2 7110
            2.6.36  Disjoint union   cdju 7112
                  2.6.36.1  Disjoint union   cdju 7112
                  ​*2.6.36.2  Left and right injections of a disjoint union   cinl 7120
                  2.6.36.3  Universal property of the disjoint union   djuss 7145
                  2.6.36.4  Dominance and equinumerosity properties of disjoint union   djudom 7168
                  2.6.36.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7177
                  2.6.36.6  Countable sets   0ct 7182
            ​*2.6.37  The one-point compactification of the natural numbers   xnninf 7194
            2.6.38  Omniscient sets   comni 7209
            2.6.39  Markov's principle   cmarkov 7226
            2.6.40  Weakly omniscient sets   cwomni 7238
            2.6.41  Cardinal numbers   ccrd 7255
            2.6.42  Axiom of Choice equivalents   wac 7288
            2.6.43  Cardinal number arithmetic   endjudisj 7293
            2.6.44  Ordinal trichotomy   exmidontriimlem1 7304
            2.6.45  Excluded middle and the power set of a singleton   pw1on 7309
            2.6.46  Apartness relations   wap 7330
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7345
​​*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 7356
            4.1.2  Final derivation of real and complex number postulates   axcnex 7943
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 7987
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 8020
            4.2.2  Infinity and the extended real number system   cpnf 8075
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8110
            4.2.4  Ordering on reals   lttr 8117
            4.2.5  Initial properties of the complex numbers   mul12 8172
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8201
            4.3.2  Subtraction   cmin 8214
            4.3.3  Multiplication   kcnktkm1cn 8426
            4.3.4  Ordering on reals (cont.)   ltadd2 8463
            4.3.5  Real Apartness   creap 8618
            4.3.6  Complex Apartness   cap 8625
            4.3.7  Reciprocals   recextlem1 8695
            4.3.8  Division   cdiv 8716
            4.3.9  Ordering on reals (cont.)   ltp1 8888
            4.3.10  Suprema   lbreu 8989
            4.3.11  Imaginary and complex number properties   crap0 9002
            4.3.12  Function operation analogue theorems   ofnegsub 9006
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 9007
            4.4.2  Principle of mathematical induction   nnind 9023
            ​*4.4.3  Decimal representation of numbers   c2 9058
            ​*4.4.4  Some properties of specific numbers   neg1cn 9112
            4.4.5  Simple number properties   halfcl 9234
            4.4.6  The Archimedean property   arch 9263
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9266
            ​*4.4.8  Extended nonnegative integers   cxnn0 9329
            4.4.9  Integers (as a subset of complex numbers)   cz 9343
            4.4.10  Decimal arithmetic   cdc 9474
            4.4.11  Upper sets of integers   cuz 9618
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9710
            4.4.13  Complex numbers as pairs of reals   cnref1o 9742
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9745
            4.5.2  Infinity and the extended real number system (cont.)   cxne 9861
            4.5.3  Real number intervals   cioo 9980
            4.5.4  Finite intervals of integers   cfz 10100
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10204
            4.5.6  Half-open integer ranges   cfzo 10234
            4.5.7  Rational numbers (cont.)   qtri3or 10347
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10375
            4.6.2  The modulo (remainder) operation   cmo 10431
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10508
            4.6.4  Strong induction over upper sets of integers   uzsinds 10553
            4.6.5  The infinite sequence builder "seq"   cseq 10556
            4.6.6  Integer powers   cexp 10647
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10828
            4.6.8  Factorial function   cfa 10834
            4.6.9  The binomial coefficient operation   cbc 10856
            4.6.10  The ` # ` (set size) function   chash 10884
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 10952
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 10996
            4.8.2  Real and imaginary parts; conjugate   ccj 11021
            4.8.3  Sequence convergence   caucvgrelemrec 11161
            4.8.4  Square root; absolute value   csqrt 11178
            4.8.5  The maximum of two real numbers   maxcom 11385
            4.8.6  The minimum of two real numbers   mincom 11411
            4.8.7  The maximum of two extended reals   xrmaxleim 11426
            4.8.8  The minimum of two extended reals   xrnegiso 11444
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 11460
            4.9.2  Finite and infinite sums   csu 11535
            4.9.3  The binomial theorem   binomlem 11665
            4.9.4  Infinite sums (cont.)   isumshft 11672
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 11679
            4.9.6  Arithmetic series   arisum 11680
            4.9.7  Geometric series   expcnvap0 11684
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 11705
            4.9.9  Mertens' theorem   mertenslemub 11716
            4.9.10  Finite and infinite products   prodf 11720
                  4.9.10.1  Product sequences   prodf 11720
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 11730
                  4.9.10.3  Complex products   cprod 11732
                  4.9.10.4  Finite products   fprodseq 11765
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 11824
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 11957
            4.10.2  _e is irrational   eirraplem 11959
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 11969
            ​*5.1.2  Even and odd numbers   evenelz 12049
            5.1.3  The division algorithm   divalglemnn 12100
            5.1.4  Bit sequences   cbits 12122
            5.1.5  The greatest common divisor operator   cgcd 12145
            5.1.6  Bézout's identity   bezoutlemnewy 12188
            5.1.7  Decidable sets of integers   nnmindc 12226
            5.1.8  Algorithms   nn0seqcvgd 12234
            5.1.9  Euclid's Algorithm   eucalgval2 12246
            ​*5.1.10  The least common multiple   clcm 12253
            ​*5.1.11  Coprimality and Euclid's lemma   coprmgcdb 12281
            5.1.12  Cancellability of congruences   congr 12293
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12300
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12337
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12354
            5.2.4  Properties of the canonical representation of a rational   cnumer 12374
            5.2.5  Euler's theorem   codz 12401
            5.2.6  Arithmetic modulo a prime number   modprm1div 12441
            5.2.7  Pythagorean Triples   coprimeprodsq 12451
            5.2.8  The prime count function   cpc 12478
            5.2.9  Pocklington's theorem   prmpwdvds 12549
            5.2.10  Infinite primes theorem   infpnlem1 12553
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 12557
            5.2.12  Lagrange's four-square theorem   cgz 12563
            5.2.13  Decimal arithmetic (cont.)   dec2dvds 12605
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 12634
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 12699
            6.1.2  Slot definitions   cplusg 12780
            6.1.3  Definition of the structure product   crest 12941
            6.1.4  Definition of the structure quotient   cimas 13001
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 13055
            ​*7.1.2  Identity elements   mgmidmo 13074
            ​*7.1.3  Iterated sums in a magma   fngsum 13090
            ​*7.1.4  Semigroups   csgrp 13103
            ​*7.1.5  Definition and basic properties of monoids   cmnd 13118
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13159
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13195
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13202
            ​*7.2.2  Group multiple operation   cmg 13325
            7.2.3  Subgroups and Quotient groups   csubg 13373
            7.2.4  Elementary theory of group homomorphisms   cghm 13446
            7.2.5  Abelian groups   ccmn 13490
                  7.2.5.1  Definition and basic properties   ccmn 13490
                  7.2.5.2  Group sum operation   gsumfzreidx 13543
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 13552
            ​*7.3.2  Non-unital rings ("rngs")   crng 13564
            ​*7.3.3  Ring unity (multiplicative identity)   cur 13591
            7.3.4  Semirings   csrg 13595
            7.3.5  Definition and basic properties of unital rings   crg 13628
            7.3.6  Opposite ring   coppr 13699
            7.3.7  Divisibility   cdsr 13718
            7.3.8  Ring homomorphisms   crh 13782
            7.3.9  Nonzero rings and zero rings   cnzr 13811
            7.3.10  Local rings   clring 13822
            7.3.11  Subrings   csubrng 13829
                  7.3.11.1  Subrings of non-unital rings   csubrng 13829
                  7.3.11.2  Subrings of unital rings   csubrg 13849
            7.3.12  Left regular elements and domains   crlreg 13887
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 13912
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 13919
            7.5.2  Subspaces and spans in a left module   clss 13984
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 14065
            7.6.2  Ideals and spans   clidl 14099
            7.6.3  Two-sided ideals and quotient rings   c2idl 14131
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14166
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14167
            ​*7.7.2  Ring of integers   czring 14222
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14243
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Abstract multivariate polynomials
            8.1.1  Definition and basic properties   cmps 14293
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 14317
                  9.1.1.1  Topologies   ctop 14317
                  9.1.1.2  Topologies on sets   ctopon 14330
                  9.1.1.3  Topological spaces   ctps 14350
            9.1.2  Topological bases   ctb 14362
            9.1.3  Examples of topologies   distop 14405
            9.1.4  Closure and interior   ccld 14412
            9.1.5  Neighborhoods   cnei 14458
            9.1.6  Subspace topologies   restrcl 14487
            9.1.7  Limits and continuity in topological spaces   ccn 14505
            9.1.8  Product topologies   ctx 14572
            9.1.9  Continuous function-builders   cnmptid 14601
            9.1.10  Homeomorphisms   chmeo 14620
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 14642
            9.2.2  Basic metric space properties   cxms 14656
            9.2.3  Metric space balls   blfvalps 14705
            9.2.4  Open sets of a metric space   mopnrel 14761
            9.2.5  Continuity in metric spaces   metcnp3 14831
            9.2.6  Topology on the reals   qtopbasss 14841
            9.2.7  Topological definitions using the reals   ccncf 14890
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 14937
            10.1.2  Intermediate value theorem   ivthinclemlm 14954
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 14974
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 14974
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 15048
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 15088
            11.2.2  Properties of pi = 3.14159...   pilem1 15099
            11.2.3  The natural logarithm on complex numbers   clog 15176
            ​*11.2.4  Logarithms to an arbitrary base   clogb 15263
            11.2.5  Quartic binomial expansion   binom4 15299
      ​11.3  Basic number theory
            11.3.1  Wilson's theorem   wilthlem1 15300
            11.3.2  Number-theoretical functions   csgm 15301
            11.3.3  Perfect Number Theorem   mersenne 15317
            ​*11.3.4  Quadratic residues and the Legendre symbol   clgs 15322
            ​*11.3.5  Gauss' Lemma   gausslemma2dlem0a 15374
            11.3.6  Quadratic reciprocity   lgseisenlem1 15395
            11.3.7  All primes 4n+1 are the sum of two squares   2sqlem1 15439
​PART 12  GUIDES AND MISCELLANEA
      ​12.1  Guides (conventions, explanations, and examples)
            ​*12.1.1  Conventions   conventions 15451
            12.1.2  Definitional examples   ex-or 15452
​PART 13  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​13.1  Mathboxes for user contributions
            13.1.1  Mathbox guidelines   mathbox 15462
      ​13.2  Mathbox for BJ
            13.2.1  Propositional calculus   bj-nnsn 15463
                  ​*13.2.1.1  Stable formulas   bj-trst 15469
                  13.2.1.2  Decidable formulas   bj-trdc 15482
            13.2.2  Predicate calculus   bj-ex 15492
            13.2.3  Set theorey miscellaneous   bj-el2oss1o 15504
            ​*13.2.4  Extensionality   bj-vtoclgft 15505
            ​*13.2.5  Decidability of classes   wdcin 15523
            13.2.6  Disjoint union   djucllem 15530
            13.2.7  Miscellaneous   funmptd 15533
            ​*13.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 15542
                  *13.2.8.1  Bounded formulas   wbd 15542
                  ​*13.2.8.2  Bounded classes   wbdc 15570
            ​*13.2.9  CZF: Bounded separation   ax-bdsep 15614
                  13.2.9.1  Delta_0-classical logic   ax-bj-d0cl 15654
                  13.2.9.2  Inductive classes and the class of natural number ordinals   wind 15656
                  ​*13.2.9.3  The first three Peano postulates   bj-peano2 15669
            ​*13.2.10  CZF: Infinity   ax-infvn 15671
                  *13.2.10.1  The set of natural number ordinals   ax-infvn 15671
                  ​*13.2.10.2  Peano's fifth postulate   bdpeano5 15673
                  ​*13.2.10.3  Bounded induction and Peano's fourth postulate   findset 15675
            ​*13.2.11  CZF: Set induction   setindft 15695
                  *13.2.11.1  Set induction   setindft 15695
                  ​*13.2.11.2  Full induction   bj-findis 15709
            ​*13.2.12  CZF: Strong collection   ax-strcoll 15712
            ​*13.2.13  CZF: Subset collection   ax-sscoll 15717
            13.2.14  Real numbers   ax-ddkcomp 15719
      ​13.3  Mathbox for Jim Kingdon
            13.3.1  Propositional and predicate logic   nnnotnotr 15720
            13.3.2  Natural numbers   1dom1el 15721
            13.3.3  The power set of a singleton   pwtrufal 15728
            13.3.4  Omniscience of NN+oo   0nninf 15735
            13.3.5  Schroeder-Bernstein Theorem   exmidsbthrlem 15753
            13.3.6  Real and complex numbers   qdencn 15758
            ​*13.3.7  Analytic omniscience principles   trilpolemclim 15767
            13.3.8  Supremum and infimum   supfz 15802
            13.3.9  Circle constant   taupi 15804
      ​13.4  Mathbox for Mykola Mostovenko
      ​13.5  Mathbox for David A. Wheeler
            13.5.1  Testable propositions   dftest 15806
            ​*13.5.2  Allsome quantifier   walsi 15807