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 7257
            2.6.42  Axiom of Choice equivalents   wac 7290
            2.6.43  Cardinal number arithmetic   endjudisj 7295
            2.6.44  Ordinal trichotomy   exmidontriimlem1 7306
            2.6.45  Excluded middle and the power set of a singleton   pw1on 7311
            2.6.46  Apartness relations   wap 7332
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7347
​​*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 7358
            4.1.2  Final derivation of real and complex number postulates   axcnex 7945
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 7989
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 8022
            4.2.2  Infinity and the extended real number system   cpnf 8077
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8112
            4.2.4  Ordering on reals   lttr 8119
            4.2.5  Initial properties of the complex numbers   mul12 8174
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8203
            4.3.2  Subtraction   cmin 8216
            4.3.3  Multiplication   kcnktkm1cn 8428
            4.3.4  Ordering on reals (cont.)   ltadd2 8465
            4.3.5  Real Apartness   creap 8620
            4.3.6  Complex Apartness   cap 8627
            4.3.7  Reciprocals   recextlem1 8697
            4.3.8  Division   cdiv 8718
            4.3.9  Ordering on reals (cont.)   ltp1 8890
            4.3.10  Suprema   lbreu 8991
            4.3.11  Imaginary and complex number properties   crap0 9004
            4.3.12  Function operation analogue theorems   ofnegsub 9008
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 9009
            4.4.2  Principle of mathematical induction   nnind 9025
            ​*4.4.3  Decimal representation of numbers   c2 9060
            ​*4.4.4  Some properties of specific numbers   neg1cn 9114
            4.4.5  Simple number properties   halfcl 9236
            4.4.6  The Archimedean property   arch 9265
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9268
            ​*4.4.8  Extended nonnegative integers   cxnn0 9331
            4.4.9  Integers (as a subset of complex numbers)   cz 9345
            4.4.10  Decimal arithmetic   cdc 9476
            4.4.11  Upper sets of integers   cuz 9620
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9712
            4.4.13  Complex numbers as pairs of reals   cnref1o 9744
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9747
            4.5.2  Infinity and the extended real number system (cont.)   cxne 9863
            4.5.3  Real number intervals   cioo 9982
            4.5.4  Finite intervals of integers   cfz 10102
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10206
            4.5.6  Half-open integer ranges   cfzo 10236
            4.5.7  Rational numbers (cont.)   qtri3or 10349
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10377
            4.6.2  The modulo (remainder) operation   cmo 10433
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10510
            4.6.4  Strong induction over upper sets of integers   uzsinds 10555
            4.6.5  The infinite sequence builder "seq"   cseq 10558
            4.6.6  Integer powers   cexp 10649
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10830
            4.6.8  Factorial function   cfa 10836
            4.6.9  The binomial coefficient operation   cbc 10858
            4.6.10  The ` # ` (set size) function   chash 10886
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 10954
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 10998
            4.8.2  Real and imaginary parts; conjugate   ccj 11023
            4.8.3  Sequence convergence   caucvgrelemrec 11163
            4.8.4  Square root; absolute value   csqrt 11180
            4.8.5  The maximum of two real numbers   maxcom 11387
            4.8.6  The minimum of two real numbers   mincom 11413
            4.8.7  The maximum of two extended reals   xrmaxleim 11428
            4.8.8  The minimum of two extended reals   xrnegiso 11446
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 11462
            4.9.2  Finite and infinite sums   csu 11537
            4.9.3  The binomial theorem   binomlem 11667
            4.9.4  Infinite sums (cont.)   isumshft 11674
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 11681
            4.9.6  Arithmetic series   arisum 11682
            4.9.7  Geometric series   expcnvap0 11686
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 11707
            4.9.9  Mertens' theorem   mertenslemub 11718
            4.9.10  Finite and infinite products   prodf 11722
                  4.9.10.1  Product sequences   prodf 11722
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 11732
                  4.9.10.3  Complex products   cprod 11734
                  4.9.10.4  Finite products   fprodseq 11767
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 11826
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 11959
            4.10.2  _e is irrational   eirraplem 11961
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 11971
            ​*5.1.2  Even and odd numbers   evenelz 12051
            5.1.3  The division algorithm   divalglemnn 12102
            5.1.4  Bit sequences   cbits 12124
            5.1.5  The greatest common divisor operator   cgcd 12147
            5.1.6  Bézout's identity   bezoutlemnewy 12190
            5.1.7  Decidable sets of integers   nnmindc 12228
            5.1.8  Algorithms   nn0seqcvgd 12236
            5.1.9  Euclid's Algorithm   eucalgval2 12248
            ​*5.1.10  The least common multiple   clcm 12255
            ​*5.1.11  Coprimality and Euclid's lemma   coprmgcdb 12283
            5.1.12  Cancellability of congruences   congr 12295
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12302
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12339
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12356
            5.2.4  Properties of the canonical representation of a rational   cnumer 12376
            5.2.5  Euler's theorem   codz 12403
            5.2.6  Arithmetic modulo a prime number   modprm1div 12443
            5.2.7  Pythagorean Triples   coprimeprodsq 12453
            5.2.8  The prime count function   cpc 12480
            5.2.9  Pocklington's theorem   prmpwdvds 12551
            5.2.10  Infinite primes theorem   infpnlem1 12555
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 12559
            5.2.12  Lagrange's four-square theorem   cgz 12565
            5.2.13  Decimal arithmetic (cont.)   dec2dvds 12607
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 12636
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 12701
            6.1.2  Slot definitions   cplusg 12782
            6.1.3  Definition of the structure product   crest 12943
            6.1.4  Definition of the structure quotient   cimas 13003
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 13057
            ​*7.1.2  Identity elements   mgmidmo 13076
            ​*7.1.3  Iterated sums in a magma   fngsum 13092
            ​*7.1.4  Semigroups   csgrp 13105
            ​*7.1.5  Definition and basic properties of monoids   cmnd 13120
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13161
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13197
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13204
            ​*7.2.2  Group multiple operation   cmg 13327
            7.2.3  Subgroups and Quotient groups   csubg 13375
            7.2.4  Elementary theory of group homomorphisms   cghm 13448
            7.2.5  Abelian groups   ccmn 13492
                  7.2.5.1  Definition and basic properties   ccmn 13492
                  7.2.5.2  Group sum operation   gsumfzreidx 13545
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 13554
            ​*7.3.2  Non-unital rings ("rngs")   crng 13566
            ​*7.3.3  Ring unity (multiplicative identity)   cur 13593
            7.3.4  Semirings   csrg 13597
            7.3.5  Definition and basic properties of unital rings   crg 13630
            7.3.6  Opposite ring   coppr 13701
            7.3.7  Divisibility   cdsr 13720
            7.3.8  Ring homomorphisms   crh 13784
            7.3.9  Nonzero rings and zero rings   cnzr 13813
            7.3.10  Local rings   clring 13824
            7.3.11  Subrings   csubrng 13831
                  7.3.11.1  Subrings of non-unital rings   csubrng 13831
                  7.3.11.2  Subrings of unital rings   csubrg 13851
            7.3.12  Left regular elements and domains   crlreg 13889
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 13914
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 13921
            7.5.2  Subspaces and spans in a left module   clss 13986
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 14067
            7.6.2  Ideals and spans   clidl 14101
            7.6.3  Two-sided ideals and quotient rings   c2idl 14133
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14168
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14169
            ​*7.7.2  Ring of integers   czring 14224
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14245
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Abstract multivariate polynomials
            8.1.1  Definition and basic properties   cmps 14295
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 14319
                  9.1.1.1  Topologies   ctop 14319
                  9.1.1.2  Topologies on sets   ctopon 14332
                  9.1.1.3  Topological spaces   ctps 14352
            9.1.2  Topological bases   ctb 14364
            9.1.3  Examples of topologies   distop 14407
            9.1.4  Closure and interior   ccld 14414
            9.1.5  Neighborhoods   cnei 14460
            9.1.6  Subspace topologies   restrcl 14489
            9.1.7  Limits and continuity in topological spaces   ccn 14507
            9.1.8  Product topologies   ctx 14574
            9.1.9  Continuous function-builders   cnmptid 14603
            9.1.10  Homeomorphisms   chmeo 14622
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 14644
            9.2.2  Basic metric space properties   cxms 14658
            9.2.3  Metric space balls   blfvalps 14707
            9.2.4  Open sets of a metric space   mopnrel 14763
            9.2.5  Continuity in metric spaces   metcnp3 14833
            9.2.6  Topology on the reals   qtopbasss 14843
            9.2.7  Topological definitions using the reals   ccncf 14892
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 14939
            10.1.2  Intermediate value theorem   ivthinclemlm 14956
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 14976
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 14976
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 15050
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 15090
            11.2.2  Properties of pi = 3.14159...   pilem1 15101
            11.2.3  The natural logarithm on complex numbers   clog 15178
            ​*11.2.4  Logarithms to an arbitrary base   clogb 15265
            11.2.5  Quartic binomial expansion   binom4 15301
      ​11.3  Basic number theory
            11.3.1  Wilson's theorem   wilthlem1 15302
            11.3.2  Number-theoretical functions   csgm 15303
            11.3.3  Perfect Number Theorem   mersenne 15319
            ​*11.3.4  Quadratic residues and the Legendre symbol   clgs 15324
            ​*11.3.5  Gauss' Lemma   gausslemma2dlem0a 15376
            11.3.6  Quadratic reciprocity   lgseisenlem1 15397
            11.3.7  All primes 4n+1 are the sum of two squares   2sqlem1 15441
​PART 12  GUIDES AND MISCELLANEA
      ​12.1  Guides (conventions, explanations, and examples)
            ​*12.1.1  Conventions   conventions 15453
            12.1.2  Definitional examples   ex-or 15454
​PART 13  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​13.1  Mathboxes for user contributions
            13.1.1  Mathbox guidelines   mathbox 15464
      ​13.2  Mathbox for BJ
            13.2.1  Propositional calculus   bj-nnsn 15465
                  ​*13.2.1.1  Stable formulas   bj-trst 15471
                  13.2.1.2  Decidable formulas   bj-trdc 15484
            13.2.2  Predicate calculus   bj-ex 15494
            13.2.3  Set theorey miscellaneous   bj-el2oss1o 15506
            ​*13.2.4  Extensionality   bj-vtoclgft 15507
            ​*13.2.5  Decidability of classes   wdcin 15525
            13.2.6  Disjoint union   djucllem 15532
            13.2.7  Miscellaneous   funmptd 15535
            ​*13.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 15544
                  *13.2.8.1  Bounded formulas   wbd 15544
                  ​*13.2.8.2  Bounded classes   wbdc 15572
            ​*13.2.9  CZF: Bounded separation   ax-bdsep 15616
                  13.2.9.1  Delta_0-classical logic   ax-bj-d0cl 15656
                  13.2.9.2  Inductive classes and the class of natural number ordinals   wind 15658
                  ​*13.2.9.3  The first three Peano postulates   bj-peano2 15671
            ​*13.2.10  CZF: Infinity   ax-infvn 15673
                  *13.2.10.1  The set of natural number ordinals   ax-infvn 15673
                  ​*13.2.10.2  Peano's fifth postulate   bdpeano5 15675
                  ​*13.2.10.3  Bounded induction and Peano's fourth postulate   findset 15677
            ​*13.2.11  CZF: Set induction   setindft 15697
                  *13.2.11.1  Set induction   setindft 15697
                  ​*13.2.11.2  Full induction   bj-findis 15711
            ​*13.2.12  CZF: Strong collection   ax-strcoll 15714
            ​*13.2.13  CZF: Subset collection   ax-sscoll 15719
            13.2.14  Real numbers   ax-ddkcomp 15721
      ​13.3  Mathbox for Jim Kingdon
            13.3.1  Propositional and predicate logic   nnnotnotr 15722
            13.3.2  Natural numbers   1dom1el 15723
            13.3.3  The power set of a singleton   pwtrufal 15730
            13.3.4  Omniscience of NN+oo   0nninf 15737
            13.3.5  Schroeder-Bernstein Theorem   exmidsbthrlem 15757
            13.3.6  Real and complex numbers   qdencn 15762
            ​*13.3.7  Analytic omniscience principles   trilpolemclim 15771
            13.3.8  Supremum and infimum   supfz 15806
            13.3.9  Circle constant   taupi 15808
      ​13.4  Mathbox for Mykola Mostovenko
      ​13.5  Mathbox for David A. Wheeler
            13.5.1  Testable propositions   dftest 15810
            ​*13.5.2  Allsome quantifier   walsi 15811