TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Uniqueness and unique existence
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Dual of an order structure
      9.2  Preordered sets and directed sets
      9.3  Partially ordered sets (posets)
      9.4  Totally ordered sets (tosets)
      9.5  Lattices
      9.6  Posets, directed sets, and lattices as relations
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Rings
      10.4  Division rings and fields
      10.5  Left modules
      10.6  Vector spaces
      10.7  Subring algebras and ideals
      10.8  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  SURREAL NUMBERS
      15.1  Sign sequence representation and Alling's axioms
      15.2  Initial consequences of Alling's axioms
      15.3  Conway cut representation
      15.4  Induction and recursion
      15.5  Surreal arithmetic
      15.6  Subsystems of surreals
PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
      16.2  Tarskian Geometry
      16.3  Properties of geometries
      16.4  Geometry in Hilbert spaces
PART 17  GRAPH THEORY
      17.1  Vertices and edges
      17.2  Undirected graphs
      17.3  Walks, paths and cycles
      17.4  Eulerian paths and the Konigsberg Bridge problem
      17.5  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
      21.4  Mathbox for Jonathan Ben-Naim
      21.5  Mathbox for BTernaryTau
      21.6  Mathbox for Mario Carneiro
      21.7  Mathbox for Filip Cernatescu
      21.8  Mathbox for Paul Chapman
      21.9  Mathbox for Hongxiu Chen
      21.10  Mathbox for Adrian Ducourtial
      21.11  Mathbox for Scott Fenton
      21.12  Mathbox for Gino Giotto
      21.13  Mathbox for Jeff Hankins
      21.14  Mathbox for Anthony Hart
      21.15  Mathbox for Chen-Pang He
      21.16  Mathbox for Jeff Hoffman
      21.17  Mathbox for Matthew House
      21.18  Mathbox for Asger C. Ipsen
      21.19  Mathbox for BJ
      21.20  Mathbox for Jim Kingdon
      21.21  Mathbox for ML
      21.22  Mathbox for Wolf Lammen
      21.23  Mathbox for Brendan Leahy
      21.24  Mathbox for Jeff Madsen
      21.25  Mathbox for Giovanni Mascellani
      21.26  Mathbox for Peter Mazsa
      21.27  Mathbox for Rodolfo Medina
      21.28  Mathbox for Norm Megill
      21.29  Mathbox for metakunt
      21.30  Mathbox for Steven Nguyen
      21.31  Mathbox for Igor Ieskov
      21.32  Mathbox for OpenAI
      21.33  Mathbox for Stefan O'Rear
      21.34  Mathbox for Noam Pasman
      21.35  Mathbox for Jon Pennant
      21.36  Mathbox for Richard Penner
      21.37  Mathbox for Stanislas Polu
      21.38  Mathbox for Rohan Ridenour
      21.39  Mathbox for Steve Rodriguez
      21.40  Mathbox for Andrew Salmon
      21.41  Mathbox for Alan Sare
      21.42  Mathbox for Eric Schmidt
      21.43  Mathbox for Glauco Siliprandi
      21.44  Mathbox for Saveliy Skresanov
      21.45  Mathbox for Ender Ting
      21.46  Mathbox for Jarvin Udandy
      21.47  Mathbox for Adhemar
      21.48  Mathbox for Alexander van der Vekens
      21.49  Mathbox for Zhi Wang
      21.50  Mathbox for Emmett Weisz
      21.51  Mathbox for David A. Wheeler
      21.52  Mathbox for Kunhao Zheng

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

​​*PART 1  CLASSICAL 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  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 113
            ​*1.2.5  Logical equivalence   wb 206
            ​*1.2.6  Logical conjunction   wa 395
            ​*1.2.7  Logical disjunction   wo 847
            ​*1.2.8  Mixed connectives   jaao 956
            ​*1.2.9  The conditional operator for propositions   wif 1062
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1082
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1085
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1490
            1.2.13  Logical "xor"   wxo 1510
            1.2.14  Logical "nor"   wnor 1527
            1.2.15  True and false constants   wal 1537
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1537
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1538
                  1.2.15.3  The true constant   wtru 1540
                  1.2.15.4  The false constant   wfal 1551
            ​*1.2.16  Truth tables   truimtru 1562
                  1.2.16.1  Implication   truimtru 1562
                  1.2.16.2  Negation   nottru 1566
                  1.2.16.3  Equivalence   trubitru 1568
                  1.2.16.4  Conjunction   truantru 1572
                  1.2.16.5  Disjunction   truortru 1576
                  1.2.16.6  Alternative denial   trunantru 1580
                  1.2.16.7  Exclusive disjunction   truxortru 1584
                  1.2.16.8  Joint denial   trunortru 1588
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1592
                  1.2.17.1  Full adder: sum   whad 1592
                  1.2.17.2  Full adder: carry   wcad 1605
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1620
            ​*1.3.2  Implicational Calculus   impsingle 1626
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1640
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1657
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1668
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1674
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1693
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1697
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1712
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1735
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1748
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1767
      ​​*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1778
                  1.4.1.1  Existential quantifier   wex 1778
                  1.4.1.2  Nonfreeness predicate   wnf 1782
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1794
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1808
                  ​*1.4.3.1  The empty domain of discourse   empty 1905
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1909
            ​*1.4.5  Equality predicate (continued)   weq 1961
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1966
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2006
            1.4.8  Define proper substitution   sbjust 2062
            1.4.9  Membership predicate   wcel 2107
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2109
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2117
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2127
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2140
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2156
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2176
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2375
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2536
            1.6.2  Unique existence: the unique existential quantifier   weu 2566
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2661
            ​*1.7.2  Intuitionistic logic   axia1 2691
​​*PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      ​2.1  ZF Set Theory - start with the Axiom of Extensionality
            2.1.1  Introduce the Axiom of Extensionality   ax-ext 2706
            2.1.2  Classes   cab 2712
                  2.1.2.1  Class abstractions   cab 2712
                  ​*2.1.2.2  Class equality   df-cleq 2726
                  2.1.2.3  Class membership   df-clel 2808
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2867
            2.1.3  Class form not-free predicate   wnfc 2882
            2.1.4  Negated equality and membership   wne 2931
                  2.1.4.1  Negated equality   wne 2931
                  2.1.4.2  Negated membership   wnel 3035
            2.1.5  Restricted quantification   wral 3050
                  2.1.5.1  Restricted universal and existential quantification   wral 3050
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3361
                  2.1.5.3  Restricted class abstraction   crab 3419
            2.1.6  The universal class   cvv 3463
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3751
            2.1.8  Russell's Paradox   rru 3767
            2.1.9  Proper substitution of classes for sets   wsbc 3770
            2.1.10  Proper substitution of classes for sets into classes   csb 3879
            2.1.11  Define basic set operations and relations   cdif 3928
            2.1.12  Subclasses and subsets   df-ss 3948
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4097
                  2.1.13.1  The difference of two classes   dfdif3 4097
                  2.1.13.2  The union of two classes   elun 4133
                  2.1.13.3  The intersection of two classes   elini 4179
                  2.1.13.4  The symmetric difference of two classes   csymdif 4232
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4245
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4287
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4305
            2.1.14  The empty set   c0 4313
            ​*2.1.15  The conditional operator for classes   cif 4505
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4564
            2.1.17  Power classes   cpw 4580
            2.1.18  Unordered and ordered pairs   snjust 4605
            2.1.19  The union of a class   cuni 4887
            2.1.20  The intersection of a class   cint 4926
            2.1.21  Indexed union and intersection   ciun 4971
            2.1.22  Disjointness   wdisj 5090
            2.1.23  Binary relations   wbr 5123
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5185
            2.1.25  Functions in maps-to notation   cmpt 5205
            2.1.26  Transitive classes   wtr 5239
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5259
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5274
            2.2.3  Derive the Null Set Axiom   axnulALT 5284
            2.2.4  Theorems requiring subset and intersection existence   nalset 5293
            2.2.5  Theorems requiring empty set existence   class2set 5335
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5345
            2.3.2  Derive the Axiom of Pairing   axprlem1 5403
            2.3.3  Ordered pair theorem   opnz 5458
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5509
            2.3.5  Power class of union and intersection   pwin 5554
            2.3.6  The identity relation   cid 5557
            2.3.7  The membership relation (or epsilon relation)   cep 5563
            ​*2.3.8  Partial and total orderings   wpo 5570
            2.3.9  Founded and well-ordering relations   wfr 5614
            2.3.10  Relations   cxp 5663
            2.3.11  The Predecessor Class   cpred 6300
            2.3.12  Well-founded induction (variant)   frpomin 6340
            2.3.13  Well-ordered induction   tz6.26 6347
            2.3.14  Ordinals   word 6362
            2.3.15  Definite description binder (inverted iota)   cio 6492
            2.3.16  Functions   wfun 6535
            2.3.17  Cantor's Theorem   canth 7367
            2.3.18  Restricted iota (description binder)   crio 7369
            2.3.19  Operations   co 7413
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7607
            2.3.20  Maps-to notation   mpondm0 7655
            2.3.21  Function operation   cof 7677
            2.3.22  Proper subset relation   crpss 7724
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7737
            2.4.2  Ordinals (continued)   epweon 7777
            2.4.3  Transfinite induction   tfi 7856
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7869
            2.4.5  Peano's postulates   peano1 7892
            2.4.6  Finite induction (for finite ordinals)   find 7899
            2.4.7  Relations and functions (cont.)   dmexg 7905
            2.4.8  First and second members of an ordered pair   c1st 7994
            2.4.9  Induction on Cartesian products   frpoins3xpg 8147
            2.4.10  Ordering on Cartesian products   xpord2lem 8149
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8164
            ​*2.4.12  The support of functions   csupp 8167
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8217
            2.4.14  Function transposition   ctpos 8232
            2.4.15  Curry and uncurry   ccur 8272
            2.4.16  Undefined values   cund 8279
            2.4.17  Well-founded recursion   cfrecs 8287
            2.4.18  Well-ordered recursion   cwrecs 8318
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8361
            2.4.20  "Strong" transfinite recursion   crecs 8392
            2.4.21  Recursive definition generator   crdg 8431
            2.4.22  Finite recursion   frfnom 8457
            2.4.23  Ordinal arithmetic   c1o 8481
            2.4.24  Natural number arithmetic   nna0 8624
            2.4.25  Natural addition   cnadd 8685
            2.4.26  Equivalence relations and classes   wer 8724
            2.4.27  The mapping operation   cmap 8848
            2.4.28  Infinite Cartesian products   cixp 8919
            2.4.29  Equinumerosity   cen 8964
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9105
            2.4.31  Equinumerosity (cont.)   xpf1o 9161
            2.4.32  Finite sets   dif1enlem 9178
            2.4.33  Pigeonhole Principle   phplem1 9226
            2.4.34  Finite sets (cont.)   onomeneq 9247
            2.4.35  Finitely supported functions   cfsupp 9383
            2.4.36  Finite intersections   cfi 9432
            2.4.37  Hall's marriage theorem   marypha1lem 9455
            2.4.38  Supremum and infimum   csup 9462
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9531
            2.4.40  Hartogs function   char 9578
            2.4.41  Weak dominance   cwdom 9586
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9614
            2.5.2  Axiom of Infinity equivalents   inf0 9643
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9660
            2.6.2  Existence of omega (the set of natural numbers)   omex 9665
            2.6.3  Cantor normal form   ccnf 9683
            2.6.4  Transitive closure of a relation   cttrcl 9729
            2.6.5  Transitive closure   trcl 9750
            2.6.6  Well-Founded Induction   frmin 9771
            2.6.7  Well-Founded Recursion   frr3g 9778
            2.6.8  Rank   cr1 9784
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9907
            2.6.10  Disjoint union   cdju 9920
            2.6.11  Cardinal numbers   ccrd 9957
            2.6.12  Axiom of Choice equivalents   wac 10137
            ​*2.6.13  Cardinal number arithmetic   undjudom 10190
            2.6.14  The Ackermann bijection   ackbij2lem1 10240
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10267
            2.6.16  Eight inequivalent definitions of finite set   sornom 10299
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10438
​​*PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      ​3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
            3.1.1  Introduce the Axiom of Countable Choice   ax-cc 10457
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10468
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10481
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10516
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10568
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10596
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10604
      ​3.3  ZFC Axioms with no distinct variable requirements
      ​3.4  The Generalized Continuum Hypothesis
            3.4.1  Sets satisfying the Generalized Continuum Hypothesis   cgch 10642
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10700
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10704
            4.1.2  Weak universes   cwun 10722
            4.1.3  Tarski classes   ctsk 10770
            4.1.4  Grothendieck universes   cgru 10812
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10845
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10848
            4.2.3  Tarski map function   ctskm 10859
​​*PART 5  REAL AND COMPLEX NUMBERS
      ​5.1  Construction and axiomatization of real and complex numbers
            5.1.1  Dedekind-cut construction of real and complex numbers   cnpi 10866
            5.1.2  Final derivation of real and complex number postulates   axaddf 11167
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11193
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11218
            5.2.2  Infinity and the extended real number system   cpnf 11274
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11314
            5.2.4  Ordering on reals   lttr 11319
            5.2.5  Initial properties of the complex numbers   mul12 11408
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11461
            5.3.2  Subtraction   cmin 11474
            5.3.3  Multiplication   kcnktkm1cn 11676
            5.3.4  Ordering on reals (cont.)   gt0ne0 11710
            5.3.5  Reciprocals   ixi 11874
            5.3.6  Division   cdiv 11902
            5.3.7  Ordering on reals (cont.)   elimgt0 12087
            5.3.8  Completeness Axiom and Suprema   fimaxre 12194
            5.3.9  Imaginary and complex number properties   inelr 12238
            5.3.10  Function operation analogue theorems   ofsubeq0 12245
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12248
            5.4.2  Principle of mathematical induction   nnind 12266
            ​*5.4.3  Decimal representation of numbers   c2 12303
            ​*5.4.4  Some properties of specific numbers   neg1cn 12362
            5.4.5  Simple number properties   halfcl 12475
            5.4.6  The Archimedean property   nnunb 12505
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12509
            ​*5.4.8  Extended nonnegative integers   cxnn0 12582
            5.4.9  Integers (as a subset of complex numbers)   cz 12596
            5.4.10  Decimal arithmetic   cdc 12716
            5.4.11  Upper sets of integers   cuz 12860
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12967
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12972
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13001
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13016
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13133
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13329
            5.5.4  Real number intervals   cioo 13369
            5.5.5  Finite intervals of integers   cfz 13529
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13640
            5.5.7  Half-open integer ranges   cfzo 13676
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13812
            5.6.2  The modulo (remainder) operation   cmo 13891
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13970
            5.6.4  Strong induction over upper sets of integers   uzsinds 14010
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14013
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14024
            5.6.7  Integer powers   cexp 14084
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14288
            5.6.9  Factorial function   cfa 14294
            5.6.10  The binomial coefficient operation   cbc 14323
            5.6.11  The ` # ` (set size) function   chash 14351
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14489
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14523
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14527
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14534
            5.7.2  Last symbol of a word   clsw 14582
            5.7.3  Concatenations of words   cconcat 14590
            5.7.4  Singleton words   cs1 14615
            5.7.5  Concatenations with singleton words   ccatws1cl 14636
            5.7.6  Subwords/substrings   csubstr 14660
            5.7.7  Prefixes of a word   cpfx 14690
            5.7.8  Subwords of subwords   swrdswrdlem 14724
            5.7.9  Subwords and concatenations   pfxcctswrd 14730
            5.7.10  Subwords of concatenations   swrdccatfn 14744
            5.7.11  Splicing words (substring replacement)   csplice 14769
            5.7.12  Reversing words   creverse 14778
            5.7.13  Repeated symbol words   creps 14788
            ​*5.7.14  Cyclical shifts of words   ccsh 14808
            5.7.15  Mapping words by a function   wrdco 14852
            5.7.16  Longer string literals   cs2 14862
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14993
            5.8.2  Basic properties of closures   cleq1lem 15003
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15006
            5.8.4  Exponentiation of relations   crelexp 15040
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15076
            ​*5.8.6  Principle of transitive induction.   relexpindlem 15084
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15087
            5.9.2  Signum (sgn or sign) function   csgn 15107
            5.9.3  Real and imaginary parts; conjugate   ccj 15117
            5.9.4  Square root; absolute value   csqrt 15254
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15488
            5.10.2  Limits   cli 15502
            5.10.3  Finite and infinite sums   csu 15704
            5.10.4  The binomial theorem   binomlem 15847
            5.10.5  The inclusion/exclusion principle   incexclem 15854
            5.10.6  Infinite sums (cont.)   isumshft 15857
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15870
            5.10.8  Arithmetic series   arisum 15878
            5.10.9  Geometric series   expcnv 15882
            5.10.10  Ratio test for infinite series convergence   cvgrat 15901
            5.10.11  Mertens' theorem   mertenslem1 15902
            5.10.12  Finite and infinite products   prodf 15905
                  5.10.12.1  Product sequences   prodf 15905
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15915
                  5.10.12.3  Complex products   cprod 15921
                  5.10.12.4  Finite products   fprod 15959
                  5.10.12.5  Infinite products   iprodclim 16016
            5.10.13  Falling and Rising Factorial   cfallfac 16022
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16064
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16079
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16220
            5.11.2  _e is irrational   eirrlem 16222
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16229
            5.12.2  The reals are uncountable   rpnnen2lem1 16232
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16266
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16270
            6.1.3  The divides relation   cdvds 16272
            ​*6.1.4  Even and odd numbers   evenelz 16355
            6.1.5  The division algorithm   divalglem0 16412
            6.1.6  Bit sequences   cbits 16438
            6.1.7  The greatest common divisor operator   cgcd 16513
            6.1.8  Bézout's identity   bezoutlem1 16558
            6.1.9  Algorithms   nn0seqcvgd 16589
            6.1.10  Euclid's Algorithm   eucalgval2 16600
            ​*6.1.11  The least common multiple   clcm 16607
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16668
            6.1.13  Cancellability of congruences   congr 16683
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16690
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16730
            6.2.3  Properties of the canonical representation of a rational   cnumer 16752
            6.2.4  Euler's theorem   codz 16782
            6.2.5  Arithmetic modulo a prime number   modprm1div 16817
            6.2.6  Pythagorean Triples   coprimeprodsq 16828
            6.2.7  The prime count function   cpc 16856
            6.2.8  Pocklington's theorem   prmpwdvds 16924
            6.2.9  Infinite primes theorem   unbenlem 16928
            6.2.10  Sum of prime reciprocals   prmreclem1 16936
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16943
            6.2.12  Lagrange's four-square theorem   cgz 16949
            6.2.13  Van der Waerden's theorem   cvdwa 16985
            6.2.14  Ramsey's theorem   cram 17019
            ​*6.2.15  Primorial function   cprmo 17051
            ​*6.2.16  Prime gaps   prmgaplem1 17069
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17083
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17113
            6.2.19  Specific prime numbers   prmlem0 17125
            6.2.20  Very large primes   1259lem1 17150
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17165
                  7.1.1.1  Extensible structures as structures with components   cstr 17165
                  7.1.1.2  Substitution of components   csts 17182
                  7.1.1.3  Slots   cslot 17200
                  ​*7.1.1.4  Structure component indices   cnx 17212
                  7.1.1.5  Base sets   cbs 17229
                  7.1.1.6  Base set restrictions   cress 17252
            7.1.2  Slot definitions   cplusg 17273
            7.1.3  Definition of the structure product   crest 17436
            7.1.4  Definition of the structure quotient   cordt 17515
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17620
            7.2.2  Independent sets in a Moore system   mrisval 17644
            7.2.3  Algebraic closure systems   isacs 17665
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17678
            8.1.2  Opposite category   coppc 17725
            8.1.3  Monomorphisms and epimorphisms   cmon 17743
            8.1.4  Sections, inverses, isomorphisms   csect 17759
            ​*8.1.5  Isomorphic objects   ccic 17810
            8.1.6  Subcategories   cssc 17822
            8.1.7  Functors   cfunc 17870
            8.1.8  Full & faithful functors   cful 17920
            8.1.9  Natural transformations and the functor category   cnat 17960
            8.1.10  Initial, terminal and zero objects of a category   cinito 17997
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18069
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18091
            8.3.2  The category of categories   ccatc 18114
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18135
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18183
            8.4.2  Functor evaluation   cevlf 18224
            8.4.3  Hom functor   chof 18263
​PART 9  BASIC ORDER THEORY
      ​9.1  Dual of an order structure
      ​9.2  Preordered sets and directed sets
      ​9.3  Partially ordered sets (posets)
      ​9.4  Totally ordered sets (tosets)
      ​9.5  Lattices
            9.5.1  Lattices   clat 18445
            9.5.2  Complete lattices   ccla 18512
            9.5.3  Distributive lattices   cdlat 18534
            9.5.4  Subset order structures   cipo 18541
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18578
            9.6.2  Directed sets, nets   cdir 18608
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18619
            ​*10.1.2  Identity elements   mgmidmo 18642
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18658
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18672
            ​*10.1.5  Semigroups   csgrp 18700
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18716
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18763
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18816
            10.1.9  Free monoids   cfrmd 18829
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18850
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18900
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18920
            ​*10.2.2  Group multiple operation   cmg 19054
            10.2.3  Subgroups and Quotient groups   csubg 19107
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19187
            10.2.5  Elementary theory of group homomorphisms   cghm 19199
            10.2.6  Isomorphisms of groups   cgim 19244
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19267
            10.2.7  Group actions   cga 19276
            10.2.8  Centralizers and centers   ccntz 19302
            10.2.9  The opposite group   coppg 19332
            10.2.10  Symmetric groups   csymg 19354
                  ​*10.2.10.1  Definition and basic properties   csymg 19354
                  10.2.10.2  Cayley's theorem   cayleylem1 19398
                  10.2.10.3  Permutations fixing one element   symgfix2 19402
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19427
                  10.2.10.5  The sign of a permutation   cpsgn 19475
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19510
            10.2.12  Direct products   clsm 19620
                  10.2.12.1  Direct products (extension)   smndlsmidm 19642
            10.2.13  Free groups   cefg 19692
            10.2.14  Abelian groups   ccmn 19766
                  10.2.14.1  Definition and basic properties   ccmn 19766
                  10.2.14.2  Cyclic groups   ccyg 19863
                  10.2.14.3  Group sum operation   gsumval3a 19889
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19969
                  10.2.14.5  Internal direct products   cdprd 19981
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20053
            10.2.15  Simple groups   csimpg 20078
                  10.2.15.1  Definition and basic properties   csimpg 20078
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20092
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20105
            ​*10.3.2  Non-unital rings ("rngs")   crng 20117
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20146
            10.3.4  Semirings   csrg 20151
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20191
            10.3.5  Unital rings   crg 20198
            10.3.6  Opposite ring   coppr 20301
            10.3.7  Divisibility   cdsr 20322
            10.3.8  Ring primes   crpm 20400
            10.3.9  Homomorphisms of non-unital rings   crnghm 20402
            10.3.10  Ring homomorphisms   crh 20437
            10.3.11  Nonzero rings and zero rings   cnzr 20480
            10.3.12  Local rings   clring 20506
            10.3.13  Subrings   csubrng 20513
                  10.3.13.1  Subrings of non-unital rings   csubrng 20513
                  10.3.13.2  Subrings of unital rings   csubrg 20537
                  10.3.13.3  Subrings generated by a subset   crgspn 20578
            10.3.14  Categories of rings   crngc 20584
                  ​*10.3.14.1  The category of non-unital rings   crngc 20584
                  ​*10.3.14.2  The category of (unital) rings   cringc 20613
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20645
            10.3.15  Left regular elements and domains   crlreg 20659
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20697
            10.4.2  Sub-division rings   csdrg 20755
            10.4.3  Absolute value (abstract algebra)   cabv 20777
            10.4.4  Star rings   cstf 20806
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20826
            10.5.2  Subspaces and spans in a left module   clss 20897
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20986
            10.5.4  Subspace sum; bases for a left module   clbs 21041
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21069
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21138
            ​*10.7.2  Left ideals and spans   clidl 21178
            10.7.3  Two-sided ideals and quotient rings   c2idl 21221
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21258
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21292
            10.7.5  Principal ideal domains   cpid 21308
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21310
            ​*10.8.2  Ring of integers   czring 21419
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21454
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21472
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21549
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21556
            10.8.6  The ordered field of real numbers   crefld 21576
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21596
            10.9.2  Orthocomplements and closed subspaces   cocv 21632
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21674
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21705
            ​*11.1.2  Free modules   cfrlm 21720
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21756
            ​*11.1.4  Independent sets and families   clindf 21778
            11.1.5  Characterization of free modules   lmimlbs 21810
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21824
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21878
            11.3.2  Polynomial evaluation   ces 22044
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22080
            ​*11.3.4  Univariate polynomials   cps1 22124
            11.3.5  Univariate polynomial evaluation   ces1 22265
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22318
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22342
            ​*11.4.2  Square matrices   cmat 22359
            ​*11.4.3  The matrix algebra   matmulr 22392
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22420
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22442
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22494
            11.4.7  Replacement functions for a square matrix   cmarrep 22510
            11.4.8  Submatrices   csubma 22530
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22538
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22578
            11.5.3  The matrix adjugate/adjunct   cmadu 22586
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22607
            11.5.5  Inverse matrix   invrvald 22630
            ​*11.5.6  Cramer's rule   slesolvec 22633
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22646
            ​*11.6.2  Constant polynomial matrices   ccpmat 22657
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22716
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22746
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22780
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22803
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22821
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22847
                  12.1.1.1  Topologies   ctop 22847
                  12.1.1.2  Topologies on sets   ctopon 22864
                  12.1.1.3  Topological spaces   ctps 22886
            12.1.2  Topological bases   ctb 22899
            12.1.3  Examples of topologies   distop 22949
            12.1.4  Closure and interior   ccld 22970
            12.1.5  Neighborhoods   cnei 23051
            12.1.6  Limit points and perfect sets   clp 23088
            12.1.7  Subspace topologies   restrcl 23111
            12.1.8  Order topology   ordtbaslem 23142
            12.1.9  Limits and continuity in topological spaces   ccn 23178
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23260
            12.1.11  Compactness   ccmp 23340
            12.1.12  Bolzano-Weierstrass theorem   bwth 23364
            12.1.13  Connectedness   cconn 23365
            12.1.14  First- and second-countability   c1stc 23391
            12.1.15  Local topological properties   clly 23418
            12.1.16  Refinements   cref 23456
            12.1.17  Compactly generated spaces   ckgen 23487
            12.1.18  Product topologies   ctx 23514
            12.1.19  Continuous function-builders   cnmptid 23615
            12.1.20  Quotient maps and quotient topology   ckq 23647
            12.1.21  Homeomorphisms   chmeo 23707
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23781
            12.2.2  Filters   cfil 23799
            12.2.3  Ultrafilters   cufil 23853
            12.2.4  Filter limits   cfm 23887
            12.2.5  Extension by continuity   ccnext 24013
            12.2.6  Topological groups   ctmd 24024
            12.2.7  Infinite group sum on topological groups   ctsu 24080
            12.2.8  Topological rings, fields, vector spaces   ctrg 24110
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24154
            12.3.2  The topology induced by an uniform structure   cutop 24185
            12.3.3  Uniform Spaces   cuss 24208
            12.3.4  Uniform continuity   cucn 24229
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24240
            12.3.6  Complete uniform spaces   ccusp 24251
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24259
            12.4.2  Basic metric space properties   cxms 24272
            12.4.3  Metric space balls   blfvalps 24338
            12.4.4  Open sets of a metric space   mopnval 24393
            12.4.5  Continuity in metric spaces   metcnp3 24497
            12.4.6  The uniform structure generated by a metric   metuval 24506
            12.4.7  Examples of metric spaces   dscmet 24529
            ​*12.4.8  Normed algebraic structures   cnm 24533
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24662
            12.4.10  Topology on the reals   qtopbaslem 24715
            12.4.11  Topological definitions using the reals   cii 24837
            12.4.12  Path homotopy   chtpy 24935
            12.4.13  The fundamental group   cpco 24969
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25031
            ​*12.5.2  Subcomplex vector spaces   ccvs 25092
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25119
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25136
            12.5.5  Convergence and completeness   ccfil 25222
            12.5.6  Baire's Category Theorem   bcthlem1 25294
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25302
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25349
            12.5.8  Euclidean spaces   crrx 25353
            12.5.9  Minimizing Vector Theorem   minveclem1 25394
            12.5.10  Projection Theorem   pjthlem1 25407
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25419
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25433
            13.2.2  Lebesgue integration   cmbf 25585
                  13.2.2.1  Lesbesgue integral   cmbf 25585
                  13.2.2.2  Lesbesgue directed integral   cdit 25817
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25833
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25833
                  13.3.1.2  Results on real differentiation   dvferm1lem 25958
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26028
            14.1.2  The division algorithm for univariate polynomials   cmn1 26101
            14.1.3  Elementary properties of complex polynomials   cply 26159
            14.1.4  The division algorithm for polynomials   cquot 26268
            14.1.5  Algebraic numbers   caa 26292
            14.1.6  Liouville's approximation theorem   aalioulem1 26310
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26330
            14.2.2  Uniform convergence   culm 26355
            14.2.3  Power series   pserval 26389
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26423
            14.3.2  Properties of pi = 3.14159...   pilem1 26431
            14.3.3  Mapping of the exponential function   efgh 26519
            14.3.4  The natural logarithm on complex numbers   clog 26532
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26743
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26780
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26818
            14.3.8  Inverse trigonometric functions   casin 26841
            14.3.9  The Birthday Problem   log2ublem1 26925
            14.3.10  Areas in R^2   carea 26934
            14.3.11  More miscellaneous converging sequences   rlimcnp 26944
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26964
            14.3.13  Euler-Mascheroni constant   cem 26971
            14.3.14  Zeta function   czeta 26992
            14.3.15  Gamma function   clgam 26995
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27047
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27052
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27060
            14.4.4  Number-theoretical functions   ccht 27070
            14.4.5  Perfect Number Theorem   mersenne 27207
            14.4.6  Characters of Z/nZ   cdchr 27212
            14.4.7  Bertrand's postulate   bcctr 27255
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27274
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27336
            14.4.10  Quadratic reciprocity   lgseisenlem1 27355
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27397
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27449
            14.4.13  The Prime Number Theorem   mudivsum 27510
            14.4.14  Ostrowski's theorem   abvcxp 27595
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27620
            15.1.2  Ordering   sltsolem1 27656
            15.1.3  Birthday Function   bdayfo 27658
            15.1.4  Density   fvnobday 27659
            ​*15.1.5  Full-Eta Property   bdayimaon 27674
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27725
            15.2.2  Birthday Theorems   bdayfun 27753
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27761
            15.3.2  Zero and One   c0s 27803
            15.3.3  Cuts and Options   cmade 27817
            15.3.4  Cofinality and coinitiality   cofsslt 27888
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27906
            15.4.2  Induction and recursion on two variables   cnorec2 27917
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27928
            15.5.2  Negation and Subtraction   cnegs 27987
            15.5.3  Multiplication   cmuls 28068
            15.5.4  Division   cdivs 28149
            15.5.5  Absolute value   cabss 28197
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28210
            15.6.2  Surreal recursive sequences   cseqs 28225
            15.6.3  Natural numbers   cnn0s 28254
            15.6.4  Integers   czs 28300
            15.6.5  Dyadic fractions   c2s 28330
            15.6.6  Real numbers   creno 28361
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28417
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28421
            16.2.2  Betweenness   tgbtwntriv2 28431
            16.2.3  Dimension   tglowdim1 28444
            16.2.4  Betweenness and Congruence   tgifscgr 28452
            16.2.5  Congruence of a series of points   ccgrg 28454
            16.2.6  Motions   cismt 28476
            16.2.7  Colinearity   tglng 28490
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28516
            16.2.9  Less-than relation in geometric congruences   cleg 28526
            16.2.10  Rays   chlg 28544
            16.2.11  Lines   btwnlng1 28563
            16.2.12  Point inversions   cmir 28596
            16.2.13  Right angles   crag 28637
            16.2.14  Half-planes   islnopp 28683
            16.2.15  Midpoints and Line Mirroring   cmid 28716
            16.2.16  Congruence of angles   ccgra 28751
            16.2.17  Angle Comparisons   cinag 28779
            16.2.18  Congruence Theorems   tgsas1 28798
            16.2.19  Equilateral triangles   ceqlg 28809
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28813
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28832
            16.4.2  Geometry in Euclidean spaces   cee 28833
                  16.4.2.1  Definition of the Euclidean space   cee 28833
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28858
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 28922
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28933
            ​*17.1.2  Vertices and indexed edges   cvtx 28941
                  17.1.2.1  Definitions and basic properties   cvtx 28941
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28948
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28956
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 28982
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 28984
            17.1.3  Edges as range of the edge function   cedg 28992
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29001
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29025
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29067
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29071
            ​*17.2.5  Undirected simple graphs   cuspgr 29093
            17.2.6  Examples for graphs   usgr0e 29181
            17.2.7  Subgraphs   csubgr 29212
            17.2.8  Finite undirected simple graphs   cfusgr 29261
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29277
                  17.2.9.1  Neighbors   cnbgr 29277
                  17.2.9.2  Universal vertices   cuvtx 29330
                  17.2.9.3  Complete graphs   ccplgr 29354
            17.2.10  Vertex degree   cvtxdg 29411
            ​*17.2.11  Regular graphs   crgr 29501
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29541
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29633
            17.3.3  Trails   ctrls 29636
            17.3.4  Paths and simple paths   cpths 29658
            17.3.5  Closed walks   cclwlks 29718
            17.3.6  Circuits and cycles   ccrcts 29732
            ​*17.3.7  Walks as words   cwwlks 29773
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29873
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29916
            ​*17.3.10  Closed walks as words   cclwwlk 29928
                  17.3.10.1  Closed walks as words   cclwwlk 29928
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29971
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30034
            17.3.11  Examples for walks, trails and paths   0ewlk 30061
            17.3.12  Connected graphs   cconngr 30133
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30144
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30193
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30205
            17.5.2  The friendship theorem for small graphs   frgr1v 30218
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30229
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30246
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30347
            18.1.2  Natural deduction   natded 30350
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30351
            18.1.4  Definitional examples   ex-or 30368
            18.1.5  Other examples   aevdemo 30407
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30410
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30421
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30434
​​*PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      ​*19.1  Additional material on group theory (deprecated)
            19.1.1  Definitions and basic properties for groups   cgr 30436
            19.1.2  Abelian groups   cablo 30491
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30505
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30528
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30531
            19.3.2  Examples of normed complex vector spaces   cnnv 30624
            19.3.3  Induced metric of a normed complex vector space   imsval 30632
            19.3.4  Inner product   cdip 30647
            19.3.5  Subspaces   css 30668
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30687
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30759
            19.5.2  Examples of pre-Hilbert spaces   cncph 30766
            19.5.3  Properties of pre-Hilbert spaces   isph 30769
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30809
            19.6.2  Examples of complex Banach spaces   cnbn 30816
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30817
            19.6.4  Minimizing Vector Theorem   minvecolem1 30821
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30832
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30845
            19.7.3  Examples of complex Hilbert spaces   cnchl 30863
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30864
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30866
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30915
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 30928
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30946
            20.1.5  Vector operations   hvmulex 30958
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31026
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31033
            20.2.2  Norms   dfhnorm2 31069
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31107
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31126
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31131
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31141
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31149
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31150
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31154
            20.4.2  Closed subspaces   df-ch 31168
            20.4.3  Orthocomplements   df-oc 31199
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31255
            20.4.5  Projection theorem   pjhthlem1 31338
            20.4.6  Projectors   df-pjh 31342
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31349
            20.5.2  Projectors (cont.)   pjhtheu2 31363
            20.5.3  Hilbert lattice operations   sh0le 31387
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31488
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31530
            20.5.6  Foulis-Holland theorem   fh1 31565
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31574
            20.5.8  Orthogonal subspaces   chscllem1 31584
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31601
            20.5.10  Projectors (cont.)   pjorthi 31616
            20.5.11  Mayet's equation E_3   mayete3i 31675
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31677
            20.6.2  Zero and identity operators   df-h0op 31695
            20.6.3  Operations on Hilbert space operators   hoaddcl 31705
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31786
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31792
            20.6.6  Adjoint   df-adjh 31796
            20.6.7  Dirac bra-ket notation   df-bra 31797
            20.6.8  Positive operators   df-leop 31799
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31800
            20.6.10  Theorems about operators and functionals   nmopval 31803
            20.6.11  Riesz lemma   riesz3i 32009
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32014
            20.6.13  Quantum computation error bound theorem   unierri 32051
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32052
            20.6.15  Positive operators (cont.)   leopg 32069
            20.6.16  Projectors as operators   pjhmopi 32093
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32158
            20.7.2  Godowski's equation   golem1 32218
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32226
            20.8.2  Atoms   df-at 32285
            20.8.3  Superposition principle   superpos 32301
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32302
            20.8.5  Irreducibility   chirredlem1 32337
            20.8.6  Atoms (cont.)   atcvat3i 32343
            20.8.7  Modular symmetry   mdsymlem1 32350
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32389
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32394
            21.3.2  Predicate Calculus   sbc2iedf 32412
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32412
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32414
                  21.3.2.3  Equality   eqtrb 32421
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32423
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32425
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32434
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32436
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32438
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32440
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32443
            21.3.3  General Set Theory   dmrab 32444
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32444
                  21.3.3.2  Image Sets   abrexdomjm 32454
                  21.3.3.3  Set relations and operations - misc additions   nelun 32460
                  21.3.3.4  Singletons   elsnd 32475
                  21.3.3.5  Unordered pairs   elpreq 32476
                  21.3.3.6  Unordered triples   tpssg 32485
                  21.3.3.7  Conditional operator - misc additions   ifeqeqx 32490
                  21.3.3.8  Set union   uniinn0 32498
                  21.3.3.9  Indexed union - misc additions   cbviunf 32503
                  21.3.3.10  Indexed intersection - misc additions   iinabrex 32517
                  21.3.3.11  Disjointness - misc additions   disjnf 32518
            21.3.4  Relations and Functions   xpdisjres 32546
                  21.3.4.1  Relations - misc additions   xpdisjres 32546
                  21.3.4.2  Functions - misc additions   feq2dd 32567
                  21.3.4.3  Operations - misc additions   mpomptxf 32622
                  21.3.4.4  The mapping operation   elmaprd 32624
                  21.3.4.5  Support of a function   suppovss 32625
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32638
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32645
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32647
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32648
                  21.3.4.10  Supremum - misc additions   supssd 32656
                  21.3.4.11  Finite Sets   imafi2 32658
                  21.3.4.12  Countable Sets   snct 32660
            21.3.5  Real and Complex Numbers   creq0 32681
                  21.3.5.1  Complex operations - misc. additions   creq0 32681
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32691
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32692
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32710
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32715
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32725
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32737
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32749
                  21.3.5.9  The greatest common divisor operator - misc. additions   znumd 32754
                  21.3.5.10  Integers   nn0split01 32759
                  21.3.5.11  Decimal numbers   dfdec100 32772
            21.3.6  Real and complex functions   nexple 32773
                  21.3.6.1  Integer powers - misc. additions   nexple 32773
                  21.3.6.2  Indicator Functions   cind 32775
            ​*21.3.7  Decimal expansion   cdp2 32793
                  ​*21.3.7.1  Decimal point   cdp 32810
                  21.3.7.2  Division in the extended real number system   cxdiv 32839
            21.3.8  Words over a set - misc additions   wrdfd 32858
                  21.3.8.1  Splicing words (substring replacement)   splfv3 32883
                  21.3.8.2  Cyclic shift of words   1cshid 32884
            21.3.9  Extensible Structures   ressplusf 32888
                  21.3.9.1  Structure restriction operator   ressplusf 32888
                  21.3.9.2  The opposite group   oppgle 32891
                  21.3.9.3  Posets   ressprs 32893
                  21.3.9.4  Complete lattices   clatp0cl 32905
                  21.3.9.5  Order Theory   cmnt 32907
                  21.3.9.6  Chains   cchn 32933
                  21.3.9.7  Extended reals Structure - misc additions   ax-xrssca 32945
                  21.3.9.8  The extended nonnegative real numbers commutative monoid   xrge0base 32955
            21.3.10  Algebra   mndcld 32966
                  21.3.10.1  Monoids   mndcld 32966
                  21.3.10.2  Monoids Homomorphisms   abliso 32980
                  21.3.10.3  Groups - misc additions   grpsubcld 32984
                  21.3.10.4  Finitely supported group sums - misc additions   gsumsubg 32988
                  21.3.10.5  Group or monoid sums over words   gsumwun 33007
                  21.3.10.6  Centralizers and centers - misc additions   cntzun 33010
                  21.3.10.7  Totally ordered monoids and groups   comnd 33013
                  21.3.10.8  The symmetric group   symgfcoeu 33041
                  21.3.10.9  Transpositions   pmtridf1o 33053
                  21.3.10.10  Permutation Signs   psgnid 33056
                  21.3.10.11  Permutation cycles   ctocyc 33065
                  21.3.10.12  The Alternating Group   evpmval 33104
                  21.3.10.13  Signum in an ordered monoid   csgns 33117
                  21.3.10.14  The Archimedean property for generic ordered algebraic structures   cinftm 33122
                  21.3.10.15  Semiring left modules   cslmd 33145
                  21.3.10.16  Simple groups   prmsimpcyc 33173
                  21.3.10.17  Rings - misc additions   ringdi22 33174
                  21.3.10.18  Subrings generated by a set   elrgspnlem1 33185
                  21.3.10.19  The zero ring   irrednzr 33193
                  21.3.10.20  Localization of rings   cerl 33196
                  21.3.10.21  Integral Domains   domnmuln0rd 33217
                  21.3.10.22  Euclidean Domains   ceuf 33230
                  21.3.10.23  Division Rings   ringinveu 33236
                  21.3.10.24  The field of rational numbers   qfld 33239
                  21.3.10.25  Subfields   subsdrg 33240
                  21.3.10.26  Field of fractions   cfrac 33244
                  21.3.10.27  Field extensions generated by a set   cfldgen 33252
                  21.3.10.28  Totally ordered rings and fields   corng 33265
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33288
                  21.3.10.30  Scalar restriction operation   cresv 33290
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33305
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33306
                  21.3.10.33  The quotient map and quotient modules   qusker 33312
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33329
                  21.3.10.35  Independent sets and families   islinds5 33330
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33347
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33353
                  21.3.10.38  The quotient map   quslsm 33368
                  21.3.10.39  Ideals   lidlmcld 33382
                  21.3.10.40  Prime Ideals   cprmidl 33398
                  21.3.10.41  Maximal Ideals   cmxidl 33422
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33463
                  21.3.10.43  Prime Elements   rprmval 33479
                  21.3.10.44  Unique factorization domains   cufd 33501
                  21.3.10.45  The ring of integers   zringidom 33514
                  21.3.10.46  Univariate Polynomials   0ringmon1p 33518
                  21.3.10.47  Polynomial quotient and polynomial remainder   q1pdir 33558
                  21.3.10.48  The subring algebra   sra1r 33567
                  21.3.10.49  Division Ring Extensions   drgext0g 33575
                  21.3.10.50  Vector Spaces   lvecdimfi 33581
                  21.3.10.51  Vector Space Dimension   cldim 33584
            21.3.11  Field Extensions   cfldext 33624
                  21.3.11.1  Algebraic numbers   cirng 33670
                  21.3.11.2  Minimal polynomials   cminply 33679
                  21.3.11.3  Quadratic Field Extensions   rtelextdg2lem 33706
                  21.3.11.4  Towers of quadratic extentions   fldext2chn 33708
            ​*21.3.12  Constructible Numbers   cconstr 33709
                  21.3.12.1  Impossible constructions   2sqr3minply 33749
            21.3.13  Matrices   csmat 33751
                  21.3.13.1  Submatrices   csmat 33751
                  21.3.13.2  Matrix literals   clmat 33769
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 33781
            21.3.14  Topology   ist0cld 33791
                  21.3.14.1  Open maps   txomap 33792
                  21.3.14.2  Topology of the unit circle   qtopt1 33793
                  21.3.14.3  Refinements   reff 33797
                  21.3.14.4  Open cover refinement property   ccref 33800
                  21.3.14.5  Lindelöf spaces   cldlf 33810
                  21.3.14.6  Paracompact spaces   cpcmp 33813
                  ​*21.3.14.7  Spectrum of a ring   crspec 33820
                  21.3.14.8  Pseudometrics   cmetid 33844
                  21.3.14.9  Continuity - misc additions   hauseqcn 33856
                  21.3.14.10  Topology of the closed unit interval   elunitge0 33857
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 33861
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 33871
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 33884
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33890
                  21.3.14.15  Limits - misc additions   lmlim 33905
                  21.3.14.16  Univariate polynomials   pl1cn 33913
            21.3.15  Uniform Stuctures and Spaces   chcmp 33914
                  21.3.15.1  Hausdorff uniform completion   chcmp 33914
            21.3.16  Topology and algebraic structures   zringnm 33916
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 33916
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 33918
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33930
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33953
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 33976
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 33979
                  ​*21.3.16.7  Topological Manifolds   cmntop 33982
                  21.3.16.8  Extended sum   cesum 33987
            21.3.17  Mixed Function/Constant operation   cofc 34055
            21.3.18  Abstract measure   csiga 34068
                  21.3.18.1  Sigma-Algebra   csiga 34068
                  21.3.18.2  Generated sigma-Algebra   csigagen 34098
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34112
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34141
                  21.3.18.5  Product Sigma-Algebra   csx 34148
                  21.3.18.6  Measures   cmeas 34155
                  21.3.18.7  The counting measure   cntmeas 34186
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34189
                  21.3.18.9  The Dirac delta measure   cdde 34192
                  21.3.18.10  The 'almost everywhere' relation   cae 34197
                  21.3.18.11  Measurable functions   cmbfm 34209
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34230
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34252
            21.3.19  Integration   itgeq12dv 34287
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34287
                  21.3.19.2  Bochner integral   citgm 34288
            21.3.20  Euler's partition theorem   oddpwdc 34315
            21.3.21  Sequences defined by strong recursion   csseq 34344
            21.3.22  Fibonacci Numbers   cfib 34357
            21.3.23  Probability   cprb 34368
                  21.3.23.1  Probability Theory   cprb 34368
                  21.3.23.2  Conditional Probabilities   ccprob 34392
                  21.3.23.3  Real-valued Random Variables   crrv 34401
                  21.3.23.4  Preimage set mapping operator   corvc 34417
                  21.3.23.5  Distribution Functions   orvcelval 34430
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34434
                  21.3.23.7  Probabilities - example   coinfliplem 34440
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34447
            21.3.24  Signum (sgn or sign) function - misc. additions   sgncl 34500
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34516
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34520
            21.3.26  Descartes's rule of signs   signspval 34526
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34526
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34536
            21.3.27  Number Theory   iblidicc 34566
                  21.3.27.1  Representations of a number as sums of integers   crepr 34582
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34609
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34618
            21.3.28  Elementary Geometry   cstrkg2d 34638
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34638
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34643
            ​*21.3.29  LeftPad Project   clpad 34648
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34671
            21.4.2  Well founded induction and recursion   bnj110 34831
            21.4.3  The existence of a minimal element in certain classes   bnj69 34983
            21.4.4  Well-founded induction   bnj1204 34985
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35035
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35041
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35045
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35046
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35046
            21.5.2  ZF set theory   exdifsn 35052
                  21.5.2.1  Finitism   prcinf 35067
                  21.5.2.2  Global choice   gblacfnacd 35072
            21.5.3  Real and complex numbers   zltp1ne 35074
            21.5.4  Graph theory   lfuhgr 35082
                  21.5.4.1  Acyclic graphs   cacycgr 35106
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35123
            21.6.2  Miscellaneous stuff   quartfull 35129
            21.6.3  Derangements and the Subfactorial   deranglem 35130
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35155
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35170
            21.6.6  Retracts and sections   cretr 35181
            21.6.7  Path-connected and simply connected spaces   cpconn 35183
            21.6.8  Covering maps   ccvm 35219
            21.6.9  Normal numbers   snmlff 35293
            21.6.10  Godel-sets of formulas - part 1   cgoe 35297
            21.6.11  Godel-sets of formulas - part 2   cgon 35396
            21.6.12  Models of ZF   cgze 35410
            ​*21.6.13  Metamath formal systems   cmcn 35424
            21.6.14  Grammatical formal systems   cm0s 35549
            21.6.15  Models of formal systems   cmuv 35569
            21.6.16  Splitting fields   ccpms 35591
            21.6.17  p-adic number fields   czr 35611
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35635
            21.8.2  Miscellaneous theorems   elfzm12 35639
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35652
            21.10.2  Clone theory   ccloneop 35654
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35660
            21.11.2  Untangled classes   untelirr 35667
            21.11.3  Extra propositional calculus theorems   3jaodd 35674
            21.11.4  Misc. Useful Theorems   nepss 35677
            21.11.5  Properties of real and complex numbers   sqdivzi 35687
            21.11.6  Infinite products   iprodefisumlem 35699
            21.11.7  Factorial limits   faclimlem1 35702
            21.11.8  Greatest common divisor and divisibility   gcd32 35708
            21.11.9  Properties of relationships   dftr6 35710
            21.11.10  Properties of functions and mappings   funpsstri 35725
            21.11.11  Set induction (or epsilon induction)   setinds 35738
            21.11.12  Ordinal numbers   elpotr 35741
            21.11.13  Defined equality axioms   axextdfeq 35757
            21.11.14  Hypothesis builders   hbntg 35765
            21.11.15  Well-founded zero, successor, and limits   cwsuc 35770
            21.11.16  Quantifier-free definitions   ctxp 35790
            21.11.17  Alternate ordered pairs   caltop 35916
            21.11.18  Geometry in the Euclidean space   cofs 35942
                  21.11.18.1  Congruence properties   cofs 35942
                  21.11.18.2  Betweenness properties   btwntriv2 35972
                  21.11.18.3  Segment Transportation   ctransport 35989
                  21.11.18.4  Properties relating betweenness and congruence   cifs 35995
                  21.11.18.5  Connectivity of betweenness   btwnconn1lem1 36047
                  21.11.18.6  Segment less than or equal to   csegle 36066
                  21.11.18.7  Outside-of relationship   coutsideof 36079
                  21.11.18.8  Lines and Rays   cline2 36094
            21.11.19  Forward difference   cfwddif 36118
            21.11.20  Rank theorems   rankung 36126
            21.11.21  Hereditarily Finite Sets   chf 36132
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems.   rmoeqi 36147
                  21.12.1.1  Inference versions.   rmoeqi 36147
                  21.12.1.2  Deduction versions.   rmoeqdv 36172
            21.12.2  Change bound variables.   in-ax8 36184
                  21.12.2.1  Change bound variables and domains.   cbvralvw2 36186
                  21.12.2.2  Change bound variables, deduction versions.   cbvmodavw 36210
                  21.12.2.3  Change bound variables and domains, deduction versions.   cbvrmodavw2 36243
            21.12.3  Study of ax-mulf usage.   mpomulnzcnf 36259
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36260
            21.13.2  Basic topological facts   topbnd 36284
            21.13.3  Topology of the real numbers   ivthALT 36295
            21.13.4  Refinements   cfne 36296
            21.13.5  Neighborhood bases determine topologies   neibastop1 36319
            21.13.6  Lattice structure of topologies   topmtcl 36323
            21.13.7  Filter bases   fgmin 36330
            21.13.8  Directed sets, nets   tailfval 36332
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36343
            21.14.2  Predicate Calculus   nalfal 36363
            21.14.3  Miscellaneous single axioms   meran1 36371
            21.14.4  Connective Symmetry   negsym1 36377
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36388
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36411
            21.16.2  gdc.mm   nnssi2 36415
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunlem1 36422
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36431
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36500
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36500
                  ​*21.19.1.2  A syntactic theorem   bj-0 36502
                  21.19.1.3  Minimal implicational calculus   bj-a1k 36504
                  ​*21.19.1.4  Positive calculus   bj-syl66ib 36515
                  21.19.1.5  Implication and negation   bj-con2com 36521
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36531
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36533
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36538
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36543
            ​*21.19.2  Modal logic   bj-axdd2 36552
            ​*21.19.3  Provability logic   cprvb 36557
            ​*21.19.4  First-order logic   bj-genr 36566
                  21.19.4.1  Adding ax-gen   bj-genr 36566
                  21.19.4.2  Adding ax-4   bj-2alim 36570
                  21.19.4.3  Adding ax-5   bj-ax12wlem 36604
                  21.19.4.4  Equality and substitution   bj-ssbeq 36613
                  21.19.4.5  Adding ax-6   bj-spimvwt 36629
                  21.19.4.6  Adding ax-7   bj-cbvexw 36636
                  21.19.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36638
                  21.19.4.8  Adding ax-11   bj-alcomexcom 36640
                  21.19.4.9  Adding ax-12   axc11n11 36642
                  21.19.4.10  Nonfreeness   wnnf 36683
                  21.19.4.11  Adding ax-13   bj-axc10 36743
                  ​*21.19.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36753
                  ​*21.19.4.13  Distinct var metavariables   bj-hbaeb2 36778
                  ​*21.19.4.14  Around ~ equsal   bj-equsal1t 36782
                  ​*21.19.4.15  Some Principia Mathematica proofs   stdpc5t 36787
                  21.19.4.16  Alternate definition of substitution   bj-sbsb 36797
                  21.19.4.17  Lemmas for substitution   bj-sbf3 36799
                  21.19.4.18  Existential uniqueness   bj-eu3f 36801
                  ​*21.19.4.19  First-order logic: miscellaneous   bj-sblem1 36802
            21.19.5  Set theory   eliminable1 36819
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 36819
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 36831
                  21.19.5.3  Characterization among sets versus among classes   elelb 36857
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36859
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 36860
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36871
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 36885
                  21.19.5.8  Generalized class abstractions   bj-cgab 36893
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 36901
                  ​*21.19.5.10  Russell's paradox   bj-ru1 36903
                  21.19.5.11  Curry's paradox in set theory   currysetlem 36905
                  ​*21.19.5.12  Some disjointness results   bj-n0i 36911
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 36915
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 36923
                  ​*21.19.5.15  Tuples of classes   bj-cproj 36950
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 36985
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 36990
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37007
                  ​*21.19.5.19  Evaluation at a class   bj-evaleq 37032
                  21.19.5.20  Elementwise operations   celwise 37039
                  ​*21.19.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37041
                  21.19.5.22  Moore collections (complements)   bj-raldifsn 37060
                  21.19.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 37076
                  ​*21.19.5.24  Currying   csethom 37082
                  ​*21.19.5.25  Setting components of extensible structures   cstrset 37094
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37097
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37097
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37110
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37132
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37138
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37156
                  ​*21.19.6.6  Addition and opposite   caddcc 37197
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37201
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37203
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37209
                  21.19.6.10  Divisibility   cnnbar 37220
            ​*21.19.7  Monoids   bj-smgrpssmgm 37228
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37243
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37246
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37246
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37268
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37270
            21.19.9  Monoid of endomorphisms   cend 37273
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37279
            21.20.2  Number theory   dfgcd3 37284
            21.20.3  Real numbers   irrdifflemf 37285
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37288
            21.21.2  Cartesian exponentiation   cfinxp 37343
            21.21.3  Topology   iunctb2 37363
                  ​*21.21.3.1  Pi-base theorems   pibp16 37373
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37382
            21.22.2  Implication chains   wl-section-impchain 37406
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37424
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37428
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37453
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37455
            21.22.7  Other stuff   wl-mps 37467
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 37680
            21.24.2  Real and complex numbers; integers   filbcmb 37706
            21.24.3  Sequences and sums   sdclem2 37708
            21.24.4  Topology   subspopn 37718
            21.24.5  Metric spaces   metf1o 37721
            21.24.6  Continuous maps and homeomorphisms   constcncf 37728
            21.24.7  Boundedness   ctotbnd 37732
            21.24.8  Isometries   cismty 37764
            21.24.9  Heine-Borel Theorem   heibor1lem 37775
            21.24.10  Banach Fixed Point Theorem   bfplem1 37788
            21.24.11  Euclidean space   crrn 37791
            21.24.12  Intervals (continued)   ismrer1 37804
            21.24.13  Operation properties   cass 37808
            21.24.14  Groups and related structures   cmagm 37814
            21.24.15  Group homomorphism and isomorphism   cghomOLD 37849
            21.24.16  Rings   crngo 37860
            21.24.17  Division Rings   cdrng 37914
            21.24.18  Ring homomorphisms   crngohom 37926
            21.24.19  Commutative rings   ccm2 37955
            21.24.20  Ideals   cidl 37973
            21.24.21  Prime rings and integral domains   cprrng 38012
            21.24.22  Ideal generators   cigen 38025
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38044
            ​*21.25.2  Tseitin axioms   fald 38095
            ​*21.25.3  Equality deductions   iuneq2f 38122
            ​*21.25.4  Miscellanea   orcomdd 38133
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38140
            21.26.2  Preparatory theorems   el2v1 38183
            21.26.3  Range Cartesian product   df-xrn 38331
            21.26.4  Cosets by ` R `   df-coss 38371
            21.26.5  Relations   df-rels 38445
            21.26.6  Subset relations   df-ssr 38458
            21.26.7  Reflexivity   df-refs 38470
            21.26.8  Converse reflexivity   df-cnvrefs 38485
            21.26.9  Symmetry   df-syms 38502
            21.26.10  Reflexivity and symmetry   symrefref2 38523
            21.26.11  Transitivity   df-trs 38532
            21.26.12  Equivalence relations   df-eqvrels 38544
            21.26.13  Redundancy   df-redunds 38583
            21.26.14  Domain quotients   df-dmqss 38598
            21.26.15  Equivalence relations on domain quotients   df-ers 38623
            21.26.16  Functions   df-funss 38640
            21.26.17  Disjoints vs. converse functions   df-disjss 38663
            21.26.18  Antisymmetry   df-antisymrel 38720
            21.26.19  Partitions: disjoints on domain quotients   df-parts 38725
            21.26.20  Partition-Equivalence Theorems   disjim 38741
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 38813
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38843
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38853
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 38867
            21.28.4  Experiments with weak deduction theorem   elimhyps 38921
            21.28.5  Miscellanea   cnaddcom 38932
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38934
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39017
            21.28.8  Opposite rings and dual vector spaces   cld 39083
            21.28.9  Ortholattices and orthomodular lattices   cops 39132
            21.28.10  Atomic lattices with covering property   ccvr 39222
            21.28.11  Hilbert lattices   chlt 39310
            21.28.12  Projective geometries based on Hilbert lattices   clln 39452
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 39752
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41441
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 41927
            21.29.2  General helpful statements   rhmzrhval 41930
            21.29.3  Some gcd and lcm results   12gcd5e1 41963
            21.29.4  Least common multiple inequality theorem   3factsumint1 41981
            21.29.5  Logarithm inequalities   3exp7 42013
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42021
            21.29.7  Sticks and stones   sticksstones1 42106
            21.29.8  Continuation AKS   aks6d1c6lem1 42130
            21.29.9  Permutation results   metakunt1 42165
            21.29.10  Unused lemmas scheduled for deletion   fac2xp3 42199
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42203
            ​*21.30.2  Arithmetic theorems   c0exALT 42251
            21.30.3  Exponents and divisibility   oexpreposd 42320
            21.30.4  Trigonometry and Calculus   tanhalfpim 42348
            21.30.5  Real subtraction   cresub 42358
            21.30.6  Structures   sn-base0 42468
            ​*21.30.7  Projective spaces   cprjsp 42574
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 42607
            ​*21.30.9  Exemplar theorems   iddii 42637
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 42648
      ​21.31  Mathbox for Igor Ieskov
      ​21.32  Mathbox for OpenAI
      ​21.33  Mathbox for Stefan O'Rear
            21.33.1  Additional elementary logic and set theory   moxfr 42666
            21.33.2  Additional theory of functions   imaiinfv 42667
            21.33.3  Additional topology   elrfi 42668
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42672
            21.33.5  Algebraic closure systems   cnacs 42676
            21.33.6  Miscellanea 1. Map utilities   constmap 42687
            21.33.7  Miscellanea for polynomials   mptfcl 42694
            21.33.8  Multivariate polynomials over the integers   cmzpcl 42695
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 42727
            21.33.10  Diophantine sets 1: definitions   cdioph 42729
            21.33.11  Diophantine sets 2 miscellanea   ellz1 42741
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 42746
            21.33.13  Diophantine sets 3: construction   diophrex 42749
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 42758
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 42768
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 42775
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 42785
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 42790
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 42794
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 42796
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 42803
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 42810
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 42852
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 42864
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 42872
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 42874
            21.33.27  Ordering and induction lemmas for the integers   monotuz 42916
            21.33.28  X and Y sequences 2: Order properties   rmxypos 42922
            21.33.29  Congruential equations   congtr 42940
            21.33.30  Alternating congruential equations   acongid 42950
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 42960
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 42963
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 42980
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 42990
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 42999
            21.33.36  More equivalents of the Axiom of Choice   axac10 43008
            21.33.37  Finitely generated left modules   clfig 43042
            21.33.38  Noetherian left modules I   clnm 43050
            21.33.39  Addenda for structure powers   pwssplit4 43064
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43070
            21.33.41  Noetherian rings and left modules II   clnr 43084
            21.33.42  Hilbert's Basis Theorem   cldgis 43096
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43106
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43115
            21.33.45  Algebraic integers I   citgo 43132
            21.33.46  Endomorphism algebra   cmend 43146
            21.33.47  Cyclic groups and order   idomodle 43166
            21.33.48  Cyclotomic polynomials   ccytp 43172
            21.33.49  Miscellaneous topology   fgraphopab 43178
      ​21.34  Mathbox for Noam Pasman
      ​21.35  Mathbox for Jon Pennant
      ​21.36  Mathbox for Richard Penner
            21.36.1  Set Theory and Ordinal Numbers   uniel 43192
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43301
            21.36.3  Surreal Contributions   abeqabi 43383
            21.36.4  Short Studies   nlimsuc 43416
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43434
                  21.36.4.2  Sophisms   rp-fakeimass 43487
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43492
                  21.36.4.4  General Observations   intabssd 43494
                  21.36.4.5  Infinite Sets   pwelg 43535
                  ​*21.36.4.6  Finite intersection property   fipjust 43540
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 43549
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 43550
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43552
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 43555
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 43571
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43575
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43576
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43579
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43583
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43605
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 43606
            21.36.5  Additional statements on relations and subclasses   al3im 43622
                  21.36.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43640
                  21.36.5.2  Reflexive closures   crcl 43647
                  ​*21.36.5.3  Finite relationship composition.   relexp2 43652
                  21.36.5.4  Transitive closure of a relation   dftrcl3 43695
                  ​*21.36.5.5  Adapted from Frege   frege77d 43721
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 43741
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 43741
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 43747
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 43765
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 43804
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 43831
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 43862
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 43889
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 43907
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 43914
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43937
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43953
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 43972
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 43972
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 43998
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44105
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44122
                  *21.36.8.1  Simplicial Sets   k0004lem1 44122
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44131
                  21.37.1.1  IMO 1972 B2   wwlemuld 44131
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44148
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44170
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44171
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44176
            21.38.2  Monoid rings   cmnring 44187
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44205
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44205
                  21.38.3.2  Minimal universes   ismnu 44237
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44264
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44281
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44288
            21.39.3  Multiples   reldvds 44291
            21.39.4  Function operations   caofcan 44299
            21.39.5  Calculus   lhe4.4ex1a 44305
            21.39.6  The generalized binomial coefficient operation   cbcc 44312
            21.39.7  Binomial series   uzmptshftfval 44322
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44334
            21.40.2  Principia Mathematica * 11   2alanimi 44348
            21.40.3  Predicate Calculus   sbeqal1 44374
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44383
            21.40.5  Set Theory   elnev 44414
            21.40.6  Arithmetic   addcomgi 44432
            21.40.7  Geometry   cplusr 44433
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44455
            21.41.2  Supplementary unification deductions   bi1imp 44459
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44478
            21.41.4  What is Virtual Deduction?   wvd1 44546
            21.41.5  Virtual Deduction Theorems   df-vd1 44547
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 44795
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 44823
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 44890
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 44894
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 44901
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 44904
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 44915
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 44917
            21.42.3  Relation-preserving functions   wrelp 44920
            21.42.4  Well-founded sets   trwf 44933
            21.42.5  Absoluteness in transitive models   ralabso 44942
            21.42.6  Lemmas for showing axioms hold in models   traxext 44951
            21.42.7  The class of well-founded sets is a model for ZFC   wfaxext 44967
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 44977
            21.43.2  Functions   fnresdmss 45130
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45241
            21.43.4  Real intervals   gtnelioc 45461
            21.43.5  Finite sums   fsummulc1f 45543
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45552
            21.43.7  Limits   clim1fr1 45573
                  21.43.7.1  Inferior limit (lim inf)   clsi 45723
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 45790
            21.43.8  Trigonometry   coseq0 45836
            21.43.9  Continuous Functions   mulcncff 45842
            21.43.10  Derivatives   dvsinexp 45883
            21.43.11  Integrals   itgsin0pilem1 45922
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 45973
            21.43.13  Wallis' product for π   wallispilem1 46037
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46046
            21.43.15  Dirichlet kernel   dirkerval 46063
            21.43.16  Fourier Series   fourierdlem1 46080
            21.43.17  e is transcendental   elaa2lem 46205
            21.43.18  n-dimensional Euclidean space   rrxtopn 46256
            21.43.19  Basic measure theory   csalg 46280
                  ​*21.43.19.1  σ-Algebras   csalg 46280
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46334
                  ​*21.43.19.3  Measures   cmea 46421
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46461
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46508
                  ​*21.43.19.6  Measurable functions   csmblfn 46667
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 46822
            21.44.2  Simple groups   simpcntrab 46842
      ​21.45  Mathbox for Ender Ting
            21.45.1  Increasing sequences and subsequences   et-ltneverrefl 46843
            21.45.2  Scratchpad for number theory   evenwodadd 46860
            21.45.3  Scratchpad for math on real numbers   squeezedltsq 46861
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 46971
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 46995
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 46995
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 46999
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47000
                  21.48.1.4  Relations - extension   eubrv 47005
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47008
                  21.48.1.6  Functions - extension   fveqvfvv 47010
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47053
            21.48.3  Double restricted existential uniqueness   r19.32 47068
                  21.48.3.1  Restricted quantification (extension)   r19.32 47068
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47077
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47080
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47083
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47086
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47092
                  21.48.4.2  The universal class (extension)   nvelim 47093
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47094
                  21.48.4.4  Predicate "defined at"   dfateq12d 47096
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47102
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47148
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47178
            21.48.6  General auxiliary theorems (2)   an4com24 47238
                  21.48.6.1  Logical conjunction - extension   an4com24 47238
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47239
                  21.48.6.3  Negated membership (alternative)   cnelbr 47241
                  21.48.6.4  The empty set - extension   ralralimp 47248
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47249
                  21.48.6.6  Functions - extension   fvifeq 47250
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47262
                  21.48.6.8  Subtraction - extension   cnambpcma 47264
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47267
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47273
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47278
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47279
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47280
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47282
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47283
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47284
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47292
                  21.48.6.18  The modulo (remainder) operation - extension   fldivmod 47298
                  21.48.6.19  The infinite sequence builder "seq"   smonoord 47316
                  21.48.6.20  Finite and infinite sums - extension   fsummsndifre 47317
                  21.48.6.21  Extensible structures - extension   setsidel 47321
            ​*21.48.7  Preimages of function values   preimafvsnel 47324
            ​*21.48.8  Partitions of real intervals   ciccp 47358
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47384
            21.48.10  Words over a set (extension)   lswn0 47389
                  21.48.10.1  Last symbol of a word - extension   lswn0 47389
            21.48.11  Unordered pairs   wich 47390
                  21.48.11.1  Interchangeable setvar variables   wich 47390
                  21.48.11.2  Set of unordered pairs   sprid 47419
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47446
                  21.48.11.4  Set of proper unordered pairs   cprpr 47457
            21.48.12  Number theory (extension)   cfmtno 47472
                  ​*21.48.12.1  Fermat numbers   cfmtno 47472
                  ​*21.48.12.2  Mersenne primes   m2prm 47536
                  21.48.12.3  Proth's theorem   modexp2m1d 47557
                  21.48.12.4  Solutions of quadratic equations   quad1 47565
            ​*21.48.13  Even and odd numbers   ceven 47569
                  21.48.13.1  Definitions and basic properties   ceven 47569
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 47594
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 47603
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 47609
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 47614
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 47619
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 47633
                  21.48.13.8  Additional theorems   epoo 47648
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 47666
            21.48.14  Number theory (extension 2)   cfppr 47669
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 47669
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 47690
            21.48.15  Graph theory (extension)   cclnbgr 47763
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 47763
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 47790
                  21.48.15.3  Induced subgraphs   cisubgr 47804
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 47818
                  ​*21.48.15.5  Triangles in graphs   cgrtri 47862
                  ​*21.48.15.6  Star graphs   cstgr 47876
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 47901
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 47957
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48006
                  21.48.15.10  Walks - extension   cupwlks 48007
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48017
            21.48.16  Monoids (extension)   ovn0dmfun 48030
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48030
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48038
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48041
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48054
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48057
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48057
                  ​*21.48.17.2  Internal binary operations   cintop 48070
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48089
            21.48.18  Rings (extension)   lmod0rng 48103
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48103
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48105
                  21.48.18.3  The non-unital ring of even integers   0even 48111
                  21.48.18.4  A constructed not unital ring   cznrnglem 48133
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48137
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48161
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48208
                  21.48.19.1  Auxiliary theorems   eliunxp2 48208
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48225
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48228
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48235
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48238
                  21.48.19.6  Divisibility (extension)   invginvrid 48241
                  21.48.19.7  The support of functions (extension)   rmsupp0 48242
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48246
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48253
                  21.48.19.10  Associative algebras (extension)   assaascl0 48255
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48257
                  21.48.19.12  Univariate polynomials (examples)   linply1 48268
            21.48.20  Linear algebra (extension)   cdmatalt 48271
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48271
                  ​*21.48.20.2  Linear combinations   clinc 48279
                  ​*21.48.20.3  Linear independence   clininds 48315
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48362
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48382
            21.48.21  Complexity theory   suppdm 48385
                  21.48.21.1  Auxiliary theorems   suppdm 48385
                  21.48.21.2  The modulo (remainder) operation (extension)   mod0mul 48398
                  21.48.21.3  Even and odd integers   nn0onn0ex 48402
                  21.48.21.4  The natural logarithm on complex numbers (extension)   logcxp0 48414
                  21.48.21.5  Division of functions   cfdiv 48416
                  21.48.21.6  Upper bounds   cbigo 48426
                  21.48.21.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 48437
                  ​*21.48.21.8  The binary logarithm   fldivexpfllog2 48444
                  21.48.21.9  Binary length   cblen 48448
                  ​*21.48.21.10  Digits   cdig 48474
                  21.48.21.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 48494
                  21.48.21.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 48503
                  ​*21.48.21.13  N-ary functions   cnaryf 48505
                  ​*21.48.21.14  The Ackermann function   citco 48536
            21.48.22  Elementary geometry (extension)   fv1prop 48578
                  21.48.22.1  Auxiliary theorems   fv1prop 48578
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 48590
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 48606
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 48668
            21.49.2  Predicate calculus with equality   dtrucor3 48677
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 48677
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 48678
                  21.49.3.1  Restricted quantification   ralbidb 48678
                  21.49.3.2  The universal class   reuxfr1dd 48684
                  21.49.3.3  The empty set   ssdisjd 48685
                  21.49.3.4  Unordered and ordered pairs   vsn 48689
                  21.49.3.5  The union of a class   unilbss 48695
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 48696
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 48696
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 48697
                  21.49.5.1  Ordered pair theorem   opth1neg 48697
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 48699
                  21.49.5.3  Relations   coxp 48702
                  21.49.5.4  Functions   mof0 48705
                  21.49.5.5  Operations   fvconstr 48723
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 48727
                  21.49.6.1  Relations and functions (cont.)   fonex 48727
                  21.49.6.2  Operations in maps-to notation (continued)   fmpodg 48728
                  21.49.6.3  Function transposition   resinsnlem 48730
                  21.49.6.4  Equinumerosity   fvconst0ci 48749
            21.49.7  Order sets   iccin 48753
                  21.49.7.1  Real number intervals   iccin 48753
            21.49.8  Extensible structures   slotresfo 48756
                  21.49.8.1  Basic definitions   slotresfo 48756
            21.49.9  Moore spaces   mreuniss 48757
            ​*21.49.10  Topology   clduni 48758
                  21.49.10.1  Closure and interior   clduni 48758
                  21.49.10.2  Neighborhoods   neircl 48762
                  21.49.10.3  Subspace topologies   restcls2lem 48770
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 48774
                  21.49.10.5  Topological definitions using the reals   iooii 48775
                  21.49.10.6  Separated sets   sepnsepolem1 48779
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 48788
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 48812
            21.49.12  Posets and lattices using extensible structures   lubeldm2 48813
                  21.49.12.1  Posets   lubeldm2 48813
                  21.49.12.2  Lattices   toslat 48839
                  21.49.12.3  Subset order structures   intubeu 48841
            21.49.13  Rings   elmgpcntrd 48862
                  21.49.13.1  Multiplicative Group   elmgpcntrd 48862
            21.49.14  Associative algebras   asclelbas 48863
                  21.49.14.1  Definition and basic properties   asclelbas 48863
            21.49.15  Categories   catprslem 48867
                  21.49.15.1  Categories   catprslem 48867
                  21.49.15.2  Opposite category   oppcmndclem 48873
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 48876
                  21.49.15.4  Sections, inverses, isomorphisms   isisod 48878
                  21.49.15.5  Functors   func1st2nd 48880
                  21.49.15.6  Universal property   upciclem1 48894
                  21.49.15.7  Natural transformations and the functor category   isnatd 48920
                  21.49.15.8  Product of categories   reldmxpc 48923
                  21.49.15.9  Swap functors   cswapf 48936
                  21.49.15.10  Transposed curry functors   cofuswapfcl 48964
                  21.49.15.11  Constant functors   diag1 48975
                  21.49.15.12  Functor composition bifunctors   fucofulem1 48981
                  21.49.15.13  Post-composition functors   postcofval 49035
                  21.49.15.14  Pre-composition functors   precofvallem 49037
            21.49.16  Examples of categories   cthinc 49044
                  21.49.16.1  Thin categories   cthinc 49044
                  21.49.16.2  Terminal categories   ctermc 49097
                  21.49.16.3  Preordered sets as thin categories   cprstc 49152
                  21.49.16.4  Monoids as categories   cmndtc 49182
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 49200
            21.50.2  Set Recursion   csetrecs 49210
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 49210
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 49226
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 49236
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 49245
            ​*21.51.2  Greater than, greater than or equal to.   cge-real 49247
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 49257
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 49268
            ​*21.51.5  Identities for "if"   ifnmfalse 49290
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 49291
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 49292
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 49294
            ​*21.51.9  Algebra helpers   mvlraddi 49298
            ​*21.51.10  Algebra helper examples   i2linesi 49305
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 49307
            ​*21.51.12  Allsome quantifier   walsi 49313
            ​*21.51.13  Miscellaneous   5m4e1 49324
            21.51.14  Theorems about algebraic numbers   aacllem 49328
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 49329