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Theorem List for Metamath Proof Explorer - 27101-27200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrmxdioph 27101 X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Xrm Dioph

Theoremjm3.1lem1 27102 Lemma for jm3.1 27105. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremjm3.1lem2 27103 Lemma for jm3.1 27105. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremjm3.1lem3 27104 Lemma for jm3.1 27105. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Yrm

Theoremjm3.1 27105 Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm Xrm Yrm

Theoremexpdiophlem1 27106* Lemma for expdioph 27108. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Yrm Yrm Xrm

Theoremexpdiophlem2 27107 Lemma for expdioph 27108. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph

Theoremexpdioph 27108 The exponential function is Diophantine. This result completes and encapsulates our development using Pell equation solution sequences and is sometimes regarded as Matiyasevich's theorem properly. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph

19.16.38  Uncategorized stuff not associated with a major project

Theoremsetindtr 27109* Epsilon induction for sets contained in a transitive set. If we are allowed to assume Infinity, then all sets have a transitive closure and this reduces to setind 7676; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremsetindtrs 27110* Epsilon induction scheme without Infinity. See comments at setindtr 27109. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3lem1 27111* Lemma for dford3 27113. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3lem2 27112* Lemma for dford3 27113. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3 27113* Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford4 27114* dford3 27113 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremwopprc 27115 Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremrpnnen3lem 27116* Lemma for rpnnen3 27117. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremrpnnen3 27117 Dedekind cut injection of into . (Contributed by Stefan O'Rear, 18-Jan-2015.)

19.16.39  More equivalents of the Axiom of Choice

Theoremaxac10 27118 Characterization of choice similar to dffin1-5 8273. (Contributed by Stefan O'Rear, 6-Jan-2015.)

Theoremharinf 27119 The Hartogs number of an infinite set is at least . MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
har

Theoremwdom2d2 27120* Deduction for weak dominance by a cross product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
*

Theoremttac 27121 Tarski's theorem about choice: infxpidm 8442 is equivalent to ax-ac 8344. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
CHOICE

Theorempw2f1ocnv 27122* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7218, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 9-Jul-2015.)

Theorempw2f1o2 27123* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7218, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theorempw2f1o2val 27124* Function value of the pw2f1o2 27123 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Theorempw2f1o2val2 27125* Membership in a mapped set under the pw2f1o2 27123 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Theoremsoeq12d 27126 Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfreq12d 27127 Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremweeq12d 27128 Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremlimsuc2 27129 Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremwepwsolem 27130* Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremwepwso 27131* A well-ordering induces a strict ordering on the power set. EDITORIAL: when well-orderings are set like, this can be strengthened to remove (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoreminisegn0 27132 Non-emptyness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremdnnumch1 27133* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 7916 (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch2 27134* Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch3lem 27135* Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch3 27136* Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnwech 27137* Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremfnwe2val 27138* Lemma for fnwe2 27142. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2lem1 27139* Lemma for fnwe2 27142. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2lem2 27140* Lemma for fnwe2 27142. An element which is in a minimal fiber and minimal within its fiber is minimal globally; thus is well-founded. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2lem3 27141* Lemma for fnwe2 27142. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2 27142* A well-ordering can be constructed on a partitioned set by patching together well-orderings on each partition using a well-ordering on the partitions themselves. Similar to fnwe 6465 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremaomclem1 27143* Lemma for dfac11 27151. This is the beginning of the proof that multiple choice is equivalent to choice. Our goal is to construct, by transfinite recursion, a well-ordering of . In what follows, is the index of the rank we wish to well-order, is the collection of well-orderings constructed so far, is the set of ordinal indexes of constructed ranks i.e. the next rank to construct, and is a postulated multiple-choice function.

Successor case 1, define a simple ordering from the well-ordered predecessor. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremaomclem2 27144* Lemma for dfac11 27151. Successor case 2, a choice function for subsets of . (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremaomclem3 27145* Lemma for dfac11 27151. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
recs

Theoremaomclem4 27146* Lemma for dfac11 27151. Limit case. Patch together well-orderings constructed so far using fnwe2 27142 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremaomclem5 27147* Lemma for dfac11 27151. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs

Theoremaomclem6 27148* Lemma for dfac11 27151. Transfinite induction, close over . (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs                             recs

Theoremaomclem7 27149* Lemma for dfac11 27151. is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs                             recs

Theoremaomclem8 27150* Lemma for dfac11 27151. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremdfac11 27151* The right-hand side of this theorem (compare with ac4 8360), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 7563, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do.

This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it.

A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.)

CHOICE

Theoremkelac1 27152* Kelley's choice, basic form: if a collection of sets can be cast as closed sets in the factors of a topology, and there is a definable element in each topology (which need not be in the closed set - if it were this would be trivial), then compactness (via finite intersection) guarantees that the final product is nonempty. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremkelac2lem 27153 Lemma for kelac2 27154 and dfac21 27155: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremkelac2 27154* Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremdfac21 27155 Tychonoff's theorem is a choice equivalent. Definition AC21 of Schechter p. 461. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 27-Aug-2015.)
CHOICE

19.16.40  Finitely generated left modules

Syntaxclfig 27156 Extend class notation with the class of finitely generated left modules.
LFinGen

Definitiondf-lfig 27157 Define the class of finitely generated left modules. Finite generation of subspaces can be intepreted using ↾s. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen

Theoremislmodfg 27158* Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen

Theoremislssfg 27159* Property of a finitely generated left (sub-)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s                      LFinGen

Theoremislssfg2 27160* Property of a finitely generated left (sub-)module, with a relaxed constraint on the spanning vectors. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s                             LFinGen

Theoremislssfgi 27161 Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LFinGen

Theoremfglmod 27162 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen

Theoremlsmfgcl 27163 The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s        s                             LFinGen       LFinGen       LFinGen

19.16.41  Noetherian left modules I

Syntaxclnm 27164 Extend class notation with the class of Noetherian left modules.
LNoeM

Definitiondf-lnm 27165* A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM s LFinGen

Theoremislnm 27166* Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM s LFinGen

Theoremislnm2 27167* Property of being a Noetherian left module with finite generation expanded in terms of spans. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeM

Theoremlnmlmod 27168 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM

Theoremlnmlssfg 27169 A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s        LNoeM LFinGen

Theoremlnmlsslnm 27170 All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s        LNoeM LNoeM

Theoremlnmfg 27171 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM LFinGen

Theoremkercvrlsm 27172 The domain of a linear function is the subspace sum of the kernel and any subspace which covers the range. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
LMHom

Theoremlmhmfgima 27173 A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s               LFinGen              LMHom        LFinGen

Theoremlnmepi 27174 Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LMHom LNoeM LNoeM

Theoremlmhmfgsplit 27175 If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
s        s        LMHom LFinGen LFinGen LFinGen

Theoremlmhmlnmsplit 27176 If the kernel and range of a homomorphism of left modules are Noetherian, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 12-Jun-2015.)
s        s        LMHom LNoeM LNoeM LNoeM

Theoremlnmlmic 27177 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚 LNoeM LNoeM

Theorempwssplit0 27178* Splitting for structure powers, part 0: restriction is a function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s

Theorempwssplit1 27179* Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s

Theorempwssplit2 27180* Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s

Theorempwssplit3 27181* Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s                             LMHom

Theorempwssplit4 27182* Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.)
s                                    s        s        s        LMIso

Theoremfilnm 27183 Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeM

Theorempwslnmlem0 27184 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LNoeM

Theorempwslnmlem1 27185* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LNoeM LNoeM

Theorempwslnmlem2 27186 A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
s        s        s                      LNoeM       LNoeM       LNoeM

Theorempwslnm 27187 Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LNoeM LNoeM

19.16.43  Direct sum of left modules

Syntaxcdsmm 27188 Class of module direct sum generator.
m

Definitiondf-dsmm 27189* The direct sum of a family of Abelian groups or left modules is the induced group structure on finite linear combinations of elements, here represented as functions with finite support. (Contributed by Stefan O'Rear, 7-Jan-2015.)
m ss

Theoremreldmdsmm 27190 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
m

Theoremdsmmval 27191* Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
s        m ss

Theoremdsmmbase 27192* Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
s        m

Theoremdsmmval2 27193 Self-referential definition of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
m        m ss

Theoremdsmmbas2 27194* Base set of the direct sum module using the fndmin 5840 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
s              m

Theoremdsmmfi 27195 For finite products, the direct sum is just the module product. (Contributed by Stefan O'Rear, 1-Feb-2015.)
m s

Theoremdsmmelbas 27196* Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s       m

Theoremdsmm0cl 27197 The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s       m

Theoremdsmmacl 27198 The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s       m

Theoremprdsinvgd2 27199 Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s

Theoremdsmmsubg 27200 The finite hull of a product of groups is additionally closed under negation and thus is a subgroup of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s       m                             SubGrp

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