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Theorem List for Metamath Proof Explorer - 27201-27300   *Has distinct variable group(s)
TypeLabelDescription
Statement

18.17.36  X and Y sequences 4: Diophantine representability of Y

Theoremjm2.27a 27201 Lemma for jm2.27 27204. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Xrm        Yrm               Xrm        Yrm               Xrm        Yrm        Yrm

Theoremjm2.27b 27202 Lemma for jm2.27 27204. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm

Theoremjm2.27c 27203 Lemma for jm2.27 27204. Forward direction with substitutions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm        Xrm        Yrm        Yrm        Xrm               Yrm        Xrm

Theoremjm2.27 27204* Lemma 2.27 of [JonesMatijasevic] p. 697; rmY is a diophantine relation. 0 was excluded from the range of B and the lower limit of G was imposed because the source proof does not seem to work otherwise; quite possible I'm just missing something. The source proof uses both i and I; i has been changed to j to avoid collision. This theorem is basically nothing but substitution instances, all the work is done in jm2.27a 27201 and jm2.27c 27203. Once Diophantine relations have been defined, the content of the theorem is "rmY is Diophantine" (Contributed by Stefan O'Rear, 4-Oct-2014.)
Yrm

Theoremjm2.27dlem1 27205* Lemma for rmydioph 27210. Subsitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem2 27206 Lemma for rmydioph 27210. This theorem is used along with the next three to efficiently infer steps like . (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem3 27207 Lemma for rmydioph 27210. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem4 27208 Lemma for rmydioph 27210. Infer -hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremjm2.27dlem5 27209 Lemma for rmydioph 27210. Used with sselii 3190 to infer membership of midpoints of range; jm2.27dlem2 27206 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)

Theoremrmydioph 27210 jm2.27 27204 restated in terms of Diophantine sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Yrm Dioph

18.17.37  X and Y sequences 5: Diophantine representability of X, ^, _C

Theoremrmxdiophlem 27211* X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.)
Xrm Yrm

Theoremrmxdioph 27212 X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Xrm Dioph

Theoremjm3.1lem1 27213 Lemma for jm3.1 27216. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremjm3.1lem2 27214 Lemma for jm3.1 27216. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm

Theoremjm3.1lem3 27215 Lemma for jm3.1 27216. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Yrm

Theoremjm3.1 27216 Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Yrm Xrm Yrm

Theoremexpdiophlem1 27217* Lemma for expdioph 27219. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Yrm Yrm Xrm

Theoremexpdiophlem2 27218 Lemma for expdioph 27219. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
Dioph

Theoremexpdioph 27219 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

18.17.38  Uncategorized stuff not associated with a major project

Theoremsetindtr 27220* 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 7435; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremsetindtrs 27221* Epsilon induction scheme without Infinity. See comments at setindtr 27220. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3lem1 27222* Lemma for dford3 27224. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3lem2 27223* Lemma for dford3 27224. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford3 27224* Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremdford4 27225* dford3 27224 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremwopprc 27226 Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)

Theoremrpnnen3lem 27227* Lemma for rpnnen3 27228. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremrpnnen3 27228 Dedekind cut injection of into . (Contributed by Stefan O'Rear, 18-Jan-2015.)

18.17.39  More equivalents of the Axiom of Choice

Theoremaxac10 27229 Characterization of choice similar to dffin1-5 8030. (Contributed by Stefan O'Rear, 6-Jan-2015.)

Theoremharinf 27230 The Hartogs number of an infinite set is at least . MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
har

Theoremwdom2d2 27231* Deduction for weak dominance by a cross product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
*

Theoremttac 27232 Tarski's theorem about choice: infxpidm 8200 is equivalent to ax-ac 8101. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
CHOICE

Theorempw2f1ocnv 27233* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 6985, 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 27234* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 6985, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theorempw2f1o2val 27235* Function value of the pw2f1o2 27234 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Theorempw2f1o2val2 27236* Membership in a mapped set under the pw2f1o2 27234 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Theoremsoeq12d 27237 Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfreq12d 27238 Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremweeq12d 27239 Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremlimsuc2 27240 Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremwepwsolem 27241* Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremwepwso 27242* 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 27243 Non-emptyness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremdnnumch1 27244* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 7673 (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch2 27245* Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch3lem 27246* Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch3 27247* Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnwech 27248* Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremfnwe2val 27249* Lemma for fnwe2 27253. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2lem1 27250* Lemma for fnwe2 27253. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2lem2 27251* Lemma for fnwe2 27253. 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 27252* Lemma for fnwe2 27253. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2 27253* 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 6247 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremaomclem1 27254* Lemma for dfac11 27263. 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 27255* Lemma for dfac11 27263. Successor case 2, a choice function for subsets of . (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremaomclem3 27256* Lemma for dfac11 27263. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
recs

Theoremaomclem4 27257* Lemma for dfac11 27263. Limit case. Patch together well-orderings constructed so far using fnwe2 27253 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremaomclem5 27258* Lemma for dfac11 27263. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs

Theoremaomclem6 27259* Lemma for dfac11 27263. Transfinite induction, close over . (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs                             recs

Theoremaomclem7 27260* Lemma for dfac11 27263. is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs                             recs

Theoremsupeq123d 27261 Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremaomclem8 27262* Lemma for dfac11 27263. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremdfac11 27263* The right-hand side of this theorem (compare with ac4 8118), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 7322, 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 27264* 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 27265 Lemma for kelac2 27266 and dfac21 27267: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremkelac2 27266* Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremdfac21 27267 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

18.17.40  Finitely generated left modules

Syntaxclfig 27268 Extend class notation with the class of finitely generated left modules.
LFinGen

Definitiondf-lfig 27269 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 27270* Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen

Theoremislssfg 27271* Property of a finitely generated left (sub-)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s                      LFinGen

Theoremislssfg2 27272* 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 27273 Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LFinGen

Theoremfglmod 27274 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen

Theoremlsmfgcl 27275 The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s        s                             LFinGen       LFinGen       LFinGen

18.17.41  Noetherian left modules I

Syntaxclnm 27276 Extend class notation with the class of Noetherian left modules.
LNoeM

Definitiondf-lnm 27277* A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM s LFinGen

Theoremislnm 27278* Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM s LFinGen

Theoremislnm2 27279* 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 27280 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM

Theoremlnmlssfg 27281 A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s        LNoeM LFinGen

Theoremlnmlsslnm 27282 All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s        LNoeM LNoeM

Theoremlnmfg 27283 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM LFinGen

Theoremkercvrlsm 27284 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 27285 A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s               LFinGen              LMHom        LFinGen

Theoremlnmepi 27286 Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LMHom LNoeM LNoeM

Theoremlmhmfgsplit 27287 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 27288 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 27289 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚 LNoeM LNoeM

Theorempwssplit0 27290* Splitting for structure powers, part 0: restriction is a function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s

Theorempwssplit1 27291* 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 27292* Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s

Theorempwssplit3 27293* Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s                             LMHom

Theorempwssplit4 27294* 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 27295 Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeM

Theorempwslnmlem0 27296 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LNoeM

Theorempwslnmlem1 27297* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LNoeM LNoeM

Theorempwslnmlem2 27298 A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
s        s        s                      LNoeM       LNoeM       LNoeM

Theorempwslnm 27299 Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LNoeM LNoeM

18.17.43  Direct sum of left modules

Syntaxcdsmm 27300 Class of module direct sum generator.
m

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