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 2968
            2.1.8  Russell's Paradox   ru 2984
            2.1.9  Proper substitution of classes for sets   wsbc 2985
            2.1.10  Proper substitution of classes for sets into classes   csb 3080
            2.1.11  Define basic set operations and relations   cdif 3150
            2.1.12  Subclasses and subsets   df-ss 3166
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3269
                  2.1.13.1  The difference of two classes   dfdif3 3269
                  2.1.13.2  The union of two classes   elun 3300
                  2.1.13.3  The intersection of two classes   elin 3342
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3390
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdifxor 3425
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3439
            2.1.14  The empty set   c0 3446
            2.1.15  Conditional operator   cif 3557
            2.1.16  Power classes   cpw 3601
            2.1.17  Unordered and ordered pairs   csn 3618
            2.1.18  The union of a class   cuni 3835
            2.1.19  The intersection of a class   cint 3870
            2.1.20  Indexed union and intersection   ciun 3912
            2.1.21  Disjointness   wdisj 4006
            2.1.22  Binary relations   wbr 4029
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4089
            2.1.24  Transitive classes   wtr 4127
      ​2.2  IZF Set Theory - add the Axioms of Collection and Separation
            2.2.1  Introduce the Axiom of Collection   ax-coll 4144
            2.2.2  Introduce the Axiom of Separation   ax-sep 4147
            2.2.3  Derive the Null Set Axiom   zfnuleu 4153
            2.2.4  Theorems requiring subset and intersection existence   nalset 4159
            2.2.5  Theorems requiring empty set existence   class2seteq 4192
            2.2.6  Collection principle   bnd 4201
      ​2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4203
            2.3.2  A notation for excluded middle   wem 4223
            2.3.3  Axiom of Pairing   ax-pr 4238
            2.3.4  Ordered pair theorem   opm 4263
            2.3.5  Ordered-pair class abstractions (cont.)   opabid 4286
            2.3.6  Power class of union and intersection   pwin 4313
            2.3.7  Epsilon and identity relations   cep 4318
            ​*2.3.8  Partial and total orderings   wpo 4325
            2.3.9  Founded and set-like relations   wfrfor 4358
            2.3.10  Ordinals   word 4393
      ​2.4  IZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4464
            2.4.2  Ordinals (continued)   ordon 4518
      ​2.5  IZF Set Theory - add the Axiom of Set Induction
            2.5.1  The ZF Axiom of Foundation would imply Excluded Middle   regexmidlemm 4564
            2.5.2  Introduce the Axiom of Set Induction   ax-setind 4569
            2.5.3  Transfinite induction   tfi 4614
      ​2.6  IZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-iinf 4620
            2.6.2  The natural numbers   com 4622
            2.6.3  Peano's postulates   peano1 4626
            2.6.4  Finite induction (for finite ordinals)   find 4631
            2.6.5  The Natural Numbers (continued)   nn0suc 4636
            2.6.6  Relations   cxp 4657
            2.6.7  Definite description binder (inverted iota)   cio 5213
            2.6.8  Functions   wfun 5248
            2.6.9  Cantor's Theorem   canth 5871
            2.6.10  Restricted iota (description binder)   crio 5872
            2.6.11  Operations   co 5918
            2.6.12  Maps-to notation   elmpocl 6113
            2.6.13  Function operation   cof 6128
            2.6.14  Functions (continued)   resfunexgALT 6160
            2.6.15  First and second members of an ordered pair   c1st 6191
            ​*2.6.16  Special maps-to operations   opeliunxp2f 6291
            2.6.17  Function transposition   ctpos 6297
            2.6.18  Undefined values   pwuninel2 6335
            2.6.19  Functions on ordinals; strictly monotone ordinal functions   iunon 6337
            2.6.20  "Strong" transfinite recursion   crecs 6357
            2.6.21  Recursive definition generator   crdg 6422
            2.6.22  Finite recursion   cfrec 6443
            2.6.23  Ordinal arithmetic   c1o 6462
            2.6.24  Natural number arithmetic   nna0 6527
            2.6.25  Equivalence relations and classes   wer 6584
            2.6.26  The mapping operation   cmap 6702
            2.6.27  Infinite Cartesian products   cixp 6752
            2.6.28  Equinumerosity   cen 6792
            2.6.29  Equinumerosity (cont.)   xpf1o 6900
            2.6.30  Pigeonhole Principle   phplem1 6908
            2.6.31  Finite sets   fict 6924
            2.6.32  Schroeder-Bernstein Theorem   sbthlem1 7016
            2.6.33  Finite intersections   cfi 7027
            2.6.34  Supremum and infimum   csup 7041
            2.6.35  Ordinal isomorphism   ordiso2 7094
            2.6.36  Disjoint union   cdju 7096
                  2.6.36.1  Disjoint union   cdju 7096
                  ​*2.6.36.2  Left and right injections of a disjoint union   cinl 7104
                  2.6.36.3  Universal property of the disjoint union   djuss 7129
                  2.6.36.4  Dominance and equinumerosity properties of disjoint union   djudom 7152
                  2.6.36.5  Older definition temporarily kept for comparison, to be deleted   cdjud 7161
                  2.6.36.6  Countable sets   0ct 7166
            ​*2.6.37  The one-point compactification of the natural numbers   xnninf 7178
            2.6.38  Omniscient sets   comni 7193
            2.6.39  Markov's principle   cmarkov 7210
            2.6.40  Weakly omniscient sets   cwomni 7222
            2.6.41  Cardinal numbers   ccrd 7239
            2.6.42  Axiom of Choice equivalents   wac 7265
            2.6.43  Cardinal number arithmetic   endjudisj 7270
            2.6.44  Ordinal trichotomy   exmidontriimlem1 7281
            2.6.45  Excluded middle and the power set of a singleton   pw1on 7286
            2.6.46  Apartness relations   wap 7307
​​*PART 3  CHOICE PRINCIPLES
      ​3.1  Countable Choice and Dependent Choice
            3.1.1  Introduce Countable Choice   wacc 7322
​​*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 7332
            4.1.2  Final derivation of real and complex number postulates   axcnex 7919
            4.1.3  Real and complex number postulates restated as axioms   ax-cnex 7963
      ​4.2  Derive the basic properties from the field axioms
            4.2.1  Some deductions from the field axioms for complex numbers   cnex 7996
            4.2.2  Infinity and the extended real number system   cpnf 8051
            4.2.3  Restate the ordering postulates with extended real "less than"   axltirr 8086
            4.2.4  Ordering on reals   lttr 8093
            4.2.5  Initial properties of the complex numbers   mul12 8148
      ​4.3  Real and complex numbers - basic operations
            4.3.1  Addition   add12 8177
            4.3.2  Subtraction   cmin 8190
            4.3.3  Multiplication   kcnktkm1cn 8402
            4.3.4  Ordering on reals (cont.)   ltadd2 8438
            4.3.5  Real Apartness   creap 8593
            4.3.6  Complex Apartness   cap 8600
            4.3.7  Reciprocals   recextlem1 8670
            4.3.8  Division   cdiv 8691
            4.3.9  Ordering on reals (cont.)   ltp1 8863
            4.3.10  Suprema   lbreu 8964
            4.3.11  Imaginary and complex number properties   crap0 8977
            4.3.12  Function operation analogue theorems   ofnegsub 8981
      ​4.4  Integer sets
            4.4.1  Positive integers (as a subset of complex numbers)   cn 8982
            4.4.2  Principle of mathematical induction   nnind 8998
            ​*4.4.3  Decimal representation of numbers   c2 9033
            ​*4.4.4  Some properties of specific numbers   neg1cn 9087
            4.4.5  Simple number properties   halfcl 9208
            4.4.6  The Archimedean property   arch 9237
            4.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 9240
            ​*4.4.8  Extended nonnegative integers   cxnn0 9303
            4.4.9  Integers (as a subset of complex numbers)   cz 9317
            4.4.10  Decimal arithmetic   cdc 9448
            4.4.11  Upper sets of integers   cuz 9592
            4.4.12  Rational numbers (as a subset of complex numbers)   cq 9684
            4.4.13  Complex numbers as pairs of reals   cnref1o 9716
      ​4.5  Order sets
            4.5.1  Positive reals (as a subset of complex numbers)   crp 9719
            4.5.2  Infinity and the extended real number system (cont.)   cxne 9835
            4.5.3  Real number intervals   cioo 9954
            4.5.4  Finite intervals of integers   cfz 10074
            ​*4.5.5  Finite intervals of nonnegative integers   elfz2nn0 10178
            4.5.6  Half-open integer ranges   cfzo 10208
            4.5.7  Rational numbers (cont.)   qtri3or 10310
      ​4.6  Elementary integer functions
            4.6.1  The floor and ceiling functions   cfl 10337
            4.6.2  The modulo (remainder) operation   cmo 10393
            4.6.3  Miscellaneous theorems about integers   frec2uz0d 10470
            4.6.4  Strong induction over upper sets of integers   uzsinds 10515
            4.6.5  The infinite sequence builder "seq"   cseq 10518
            4.6.6  Integer powers   cexp 10609
            4.6.7  Ordered pair theorem for nonnegative integers   nn0le2msqd 10790
            4.6.8  Factorial function   cfa 10796
            4.6.9  The binomial coefficient operation   cbc 10818
            4.6.10  The ` # ` (set size) function   chash 10846
      ​​*4.7  Words over a set
            4.7.1  Definitions and basic theorems   cword 10914
      ​4.8  Elementary real and complex functions
            4.8.1  The "shift" operation   cshi 10958
            4.8.2  Real and imaginary parts; conjugate   ccj 10983
            4.8.3  Sequence convergence   caucvgrelemrec 11123
            4.8.4  Square root; absolute value   csqrt 11140
            4.8.5  The maximum of two real numbers   maxcom 11347
            4.8.6  The minimum of two real numbers   mincom 11372
            4.8.7  The maximum of two extended reals   xrmaxleim 11387
            4.8.8  The minimum of two extended reals   xrnegiso 11405
      ​4.9  Elementary limits and convergence
            4.9.1  Limits   cli 11421
            4.9.2  Finite and infinite sums   csu 11496
            4.9.3  The binomial theorem   binomlem 11626
            4.9.4  Infinite sums (cont.)   isumshft 11633
            4.9.5  Miscellaneous converging and diverging sequences   divcnv 11640
            4.9.6  Arithmetic series   arisum 11641
            4.9.7  Geometric series   expcnvap0 11645
            4.9.8  Ratio test for infinite series convergence   cvgratnnlembern 11666
            4.9.9  Mertens' theorem   mertenslemub 11677
            4.9.10  Finite and infinite products   prodf 11681
                  4.9.10.1  Product sequences   prodf 11681
                  4.9.10.2  Non-trivial convergence   ntrivcvgap 11691
                  4.9.10.3  Complex products   cprod 11693
                  4.9.10.4  Finite products   fprodseq 11726
      ​4.10  Elementary trigonometry
            4.10.1  The exponential, sine, and cosine functions   ce 11785
                  4.10.1.1  The circle constant (tau = 2 pi)   ctau 11918
            4.10.2  _e is irrational   eirraplem 11920
​​*PART 5  ELEMENTARY NUMBER THEORY
      ​5.1  Elementary properties of divisibility
            5.1.1  The divides relation   cdvds 11930
            ​*5.1.2  Even and odd numbers   evenelz 12008
            5.1.3  The division algorithm   divalglemnn 12059
            5.1.4  The greatest common divisor operator   cgcd 12079
            5.1.5  Bézout's identity   bezoutlemnewy 12133
            5.1.6  Decidable sets of integers   nnmindc 12171
            5.1.7  Algorithms   nn0seqcvgd 12179
            5.1.8  Euclid's Algorithm   eucalgval2 12191
            ​*5.1.9  The least common multiple   clcm 12198
            ​*5.1.10  Coprimality and Euclid's lemma   coprmgcdb 12226
            5.1.11  Cancellability of congruences   congr 12238
      ​5.2  Elementary prime number theory
            ​*5.2.1  Elementary properties   cprime 12245
            ​*5.2.2  Coprimality and Euclid's lemma (cont.)   coprm 12282
            5.2.3  Non-rationality of square root of 2   sqrt2irrlem 12299
            5.2.4  Properties of the canonical representation of a rational   cnumer 12319
            5.2.5  Euler's theorem   codz 12346
            5.2.6  Arithmetic modulo a prime number   modprm1div 12385
            5.2.7  Pythagorean Triples   coprimeprodsq 12395
            5.2.8  The prime count function   cpc 12422
            5.2.9  Pocklington's theorem   prmpwdvds 12493
            5.2.10  Infinite primes theorem   infpnlem1 12497
            5.2.11  Fundamental theorem of arithmetic   1arithlem1 12501
            5.2.12  Lagrange's four-square theorem   cgz 12507
      ​5.3  Cardinality of real and complex number subsets
            5.3.1  Countability of integers and rationals   oddennn 12549
​PART 6  BASIC STRUCTURES
      ​6.1  Extensible structures
            ​*6.1.1  Basic definitions   cstr 12614
            6.1.2  Slot definitions   cplusg 12695
            6.1.3  Definition of the structure product   crest 12850
            6.1.4  Definition of the structure quotient   cimas 12882
​PART 7  BASIC ALGEBRAIC STRUCTURES
      ​7.1  Monoids
            ​*7.1.1  Magmas   cplusf 12936
            ​*7.1.2  Identity elements   mgmidmo 12955
            ​*7.1.3  Iterated sums in a magma   fngsum 12971
            ​*7.1.4  Semigroups   csgrp 12984
            ​*7.1.5  Definition and basic properties of monoids   cmnd 12997
            7.1.6  Monoid homomorphisms and submonoids   cmhm 13029
            ​*7.1.7  Iterated sums in a monoid   gsumvallem2 13065
      ​7.2  Groups
            7.2.1  Definition and basic properties   cgrp 13072
            ​*7.2.2  Group multiple operation   cmg 13189
            7.2.3  Subgroups and Quotient groups   csubg 13237
            7.2.4  Elementary theory of group homomorphisms   cghm 13310
            7.2.5  Abelian groups   ccmn 13354
                  7.2.5.1  Definition and basic properties   ccmn 13354
                  7.2.5.2  Group sum operation   gsumfzreidx 13407
      ​7.3  Rings
            7.3.1  Multiplicative Group   cmgp 13416
            ​*7.3.2  Non-unital rings ("rngs")   crng 13428
            ​*7.3.3  Ring unity (multiplicative identity)   cur 13455
            7.3.4  Semirings   csrg 13459
            7.3.5  Definition and basic properties of unital rings   crg 13492
            7.3.6  Opposite ring   coppr 13563
            7.3.7  Divisibility   cdsr 13582
            7.3.8  Ring homomorphisms   crh 13646
            7.3.9  Nonzero rings and zero rings   cnzr 13675
            7.3.10  Local rings   clring 13686
            7.3.11  Subrings   csubrng 13693
                  7.3.11.1  Subrings of non-unital rings   csubrng 13693
                  7.3.11.2  Subrings of unital rings   csubrg 13713
            7.3.12  Left regular elements and domains   crlreg 13751
      ​7.4  Division rings and fields
            7.4.1  Ring apartness   capr 13776
      ​7.5  Left modules
            7.5.1  Definition and basic properties   clmod 13783
            7.5.2  Subspaces and spans in a left module   clss 13848
      ​7.6  Subring algebras and ideals
            7.6.1  Subring algebras   csra 13929
            7.6.2  Ideals and spans   clidl 13963
            7.6.3  Two-sided ideals and quotient rings   c2idl 13995
            7.6.4  Principal ideal rings. Divisibility in the integers   rspsn 14030
      ​7.7  The complex numbers as an algebraic extensible structure
            7.7.1  Definition and basic properties   cpsmet 14031
            ​*7.7.2  Ring of integers   czring 14078
            7.7.3  Algebraic constructions based on the complex numbers   czrh 14099
​​*PART 8  BASIC LINEAR ALGEBRA
      ​8.1  Abstract multivariate polynomials
            8.1.1  Definition and basic properties   cmps 14149
​PART 9  BASIC TOPOLOGY
      ​9.1  Topology
            ​*9.1.1  Topological spaces   ctop 14165
                  9.1.1.1  Topologies   ctop 14165
                  9.1.1.2  Topologies on sets   ctopon 14178
                  9.1.1.3  Topological spaces   ctps 14198
            9.1.2  Topological bases   ctb 14210
            9.1.3  Examples of topologies   distop 14253
            9.1.4  Closure and interior   ccld 14260
            9.1.5  Neighborhoods   cnei 14306
            9.1.6  Subspace topologies   restrcl 14335
            9.1.7  Limits and continuity in topological spaces   ccn 14353
            9.1.8  Product topologies   ctx 14420
            9.1.9  Continuous function-builders   cnmptid 14449
            9.1.10  Homeomorphisms   chmeo 14468
      ​9.2  Metric spaces
            9.2.1  Pseudometric spaces   psmetrel 14490
            9.2.2  Basic metric space properties   cxms 14504
            9.2.3  Metric space balls   blfvalps 14553
            9.2.4  Open sets of a metric space   mopnrel 14609
            9.2.5  Continuity in metric spaces   metcnp3 14679
            9.2.6  Topology on the reals   qtopbasss 14689
            9.2.7  Topological definitions using the reals   ccncf 14725
​PART 10  BASIC REAL AND COMPLEX ANALYSIS
      ​10.1  Continuity
            10.1.1  Dedekind cuts   dedekindeulemuub 14771
            10.1.2  Intermediate value theorem   ivthinclemlm 14788
      ​10.2  Derivatives
            10.2.1  Real and complex differentiation   climc 14808
                  10.2.1.1  Derivatives of functions of one complex or real variable   climc 14808
​PART 11  BASIC REAL AND COMPLEX FUNCTIONS
      ​11.1  Polynomials
            11.1.1  Elementary properties of complex polynomials   cply 14874
      ​11.2  Basic trigonometry
            11.2.1  The exponential, sine, and cosine functions (cont.)   efcn 14903
            11.2.2  Properties of pi = 3.14159...   pilem1 14914
            11.2.3  The natural logarithm on complex numbers   clog 14991
            ​*11.2.4  Logarithms to an arbitrary base   clogb 15075
            11.2.5  Quartic binomial expansion   binom4 15111
      ​11.3  Basic number theory
            11.3.1  Wilson's theorem   wilthlem1 15112
            ​*11.3.2  Quadratic residues and the Legendre symbol   clgs 15113
            ​*11.3.3  Gauss' Lemma   gausslemma2dlem0a 15165
            11.3.4  Quadratic reciprocity   lgseisenlem1 15186
            11.3.5  All primes 4n+1 are the sum of two squares   2sqlem1 15201
​PART 12  GUIDES AND MISCELLANEA
      ​12.1  Guides (conventions, explanations, and examples)
            ​*12.1.1  Conventions   conventions 15213
            12.1.2  Definitional examples   ex-or 15214
​PART 13  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​13.1  Mathboxes for user contributions
            13.1.1  Mathbox guidelines   mathbox 15224
      ​13.2  Mathbox for BJ
            13.2.1  Propositional calculus   bj-nnsn 15225
                  ​*13.2.1.1  Stable formulas   bj-trst 15231
                  13.2.1.2  Decidable formulas   bj-trdc 15244
            13.2.2  Predicate calculus   bj-ex 15254
            13.2.3  Set theorey miscellaneous   bj-el2oss1o 15266
            ​*13.2.4  Extensionality   bj-vtoclgft 15267
            ​*13.2.5  Decidability of classes   wdcin 15285
            13.2.6  Disjoint union   djucllem 15292
            13.2.7  Miscellaneous   funmptd 15295
            ​*13.2.8  Constructive Zermelo--Fraenkel set theory (CZF): Bounded formulas and classes   wbd 15304
                  *13.2.8.1  Bounded formulas   wbd 15304
                  ​*13.2.8.2  Bounded classes   wbdc 15332
            ​*13.2.9  CZF: Bounded separation   ax-bdsep 15376
                  13.2.9.1  Delta_0-classical logic   ax-bj-d0cl 15416
                  13.2.9.2  Inductive classes and the class of natural number ordinals   wind 15418
                  ​*13.2.9.3  The first three Peano postulates   bj-peano2 15431
            ​*13.2.10  CZF: Infinity   ax-infvn 15433
                  *13.2.10.1  The set of natural number ordinals   ax-infvn 15433
                  ​*13.2.10.2  Peano's fifth postulate   bdpeano5 15435
                  ​*13.2.10.3  Bounded induction and Peano's fourth postulate   findset 15437
            ​*13.2.11  CZF: Set induction   setindft 15457
                  *13.2.11.1  Set induction   setindft 15457
                  ​*13.2.11.2  Full induction   bj-findis 15471
            ​*13.2.12  CZF: Strong collection   ax-strcoll 15474
            ​*13.2.13  CZF: Subset collection   ax-sscoll 15479
            13.2.14  Real numbers   ax-ddkcomp 15481
      ​13.3  Mathbox for Jim Kingdon
            13.3.1  Propositional and predicate logic   nnnotnotr 15482
            13.3.2  Natural numbers   1dom1el 15483
            13.3.3  The power set of a singleton   pwtrufal 15488
            13.3.4  Omniscience of NN+oo   0nninf 15494
            13.3.5  Schroeder-Bernstein Theorem   exmidsbthrlem 15512
            13.3.6  Real and complex numbers   qdencn 15517
            ​*13.3.7  Analytic omniscience principles   trilpolemclim 15526
            13.3.8  Supremum and infimum   supfz 15561
            13.3.9  Circle constant   taupi 15563
      ​13.4  Mathbox for Mykola Mostovenko
      ​13.5  Mathbox for David A. Wheeler
            13.5.1  Testable propositions   dftest 15565
            ​*13.5.2  Allsome quantifier   walsi 15566