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Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempw2f1o2val2 27001* Membership in a mapped set under the pw2f1o2 26999 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Theoremsoeq12d 27002 Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfreq12d 27003 Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremweeq12d 27004 Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremlimsuc2 27005 Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremwepwsolem 27006* Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremwepwso 27007* 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 27008 Non-emptyness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremdnnumch1 27009* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 7867 (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch2 27010* Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch3lem 27011* Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnnumch3 27012* Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremdnwech 27013* Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremfnwe2val 27014* Lemma for fnwe2 27018. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2lem1 27015* Lemma for fnwe2 27018. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2lem2 27016* Lemma for fnwe2 27018. 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 27017* Lemma for fnwe2 27018. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremfnwe2 27018* 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 6421 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Theoremaomclem1 27019* Lemma for dfac11 27028. 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 27020* Lemma for dfac11 27028. Successor case 2, a choice function for subsets of . (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremaomclem3 27021* Lemma for dfac11 27028. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
recs

Theoremaomclem4 27022* Lemma for dfac11 27028. Limit case. Patch together well-orderings constructed so far using fnwe2 27018 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremaomclem5 27023* Lemma for dfac11 27028. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs

Theoremaomclem6 27024* Lemma for dfac11 27028. Transfinite induction, close over . (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs                             recs

Theoremaomclem7 27025* Lemma for dfac11 27028. is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)
recs                             recs

Theoremsupeq123d 27026 Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremaomclem8 27027* Lemma for dfac11 27028. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Theoremdfac11 27028* The right-hand side of this theorem (compare with ac4 8311), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 7516, 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 27029* 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 27030 Lemma for kelac2 27031 and dfac21 27032: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremkelac2 27031* Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremdfac21 27032 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 27033 Extend class notation with the class of finitely generated left modules.
LFinGen

Definitiondf-lfig 27034 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 27035* Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen

Theoremislssfg 27036* Property of a finitely generated left (sub-)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s                      LFinGen

Theoremislssfg2 27037* 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 27038 Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LFinGen

Theoremfglmod 27039 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen

Theoremlsmfgcl 27040 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 27041 Extend class notation with the class of Noetherian left modules.
LNoeM

Definitiondf-lnm 27042* A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM s LFinGen

Theoremislnm 27043* Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM s LFinGen

Theoremislnm2 27044* 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 27045 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM

Theoremlnmlssfg 27046 A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s        LNoeM LFinGen

Theoremlnmlsslnm 27047 All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s        LNoeM LNoeM

Theoremlnmfg 27048 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM LFinGen

Theoremkercvrlsm 27049 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 27050 A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s               LFinGen              LMHom        LFinGen

Theoremlnmepi 27051 Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LMHom LNoeM LNoeM

Theoremlmhmfgsplit 27052 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 27053 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 27054 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚 LNoeM LNoeM

19.16.42  Addenda for structure powers

Theorempwssplit0 27055* Splitting for structure powers, part 0: restriction is a function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s

Theorempwssplit1 27056* 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 27057* Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s

Theorempwssplit3 27058* Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        s                             LMHom

Theorempwssplit4 27059* 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 27060 Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeM

Theorempwslnmlem0 27061 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LNoeM

Theorempwslnmlem1 27062* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s        LNoeM LNoeM

Theorempwslnmlem2 27063 A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
s        s        s                      LNoeM       LNoeM       LNoeM

Theorempwslnm 27064 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 27065 Class of module direct sum generator.
m

Definitiondf-dsmm 27066* 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 27067 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
m

Theoremdsmmval 27068* Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
s        m ss

Theoremdsmmbase 27069* Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
s        m

Theoremdsmmval2 27070 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 27071* Base set of the direct sum module using the fndmin 5796 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
s              m

Theoremdsmmfi 27072 For finite products, the direct sum is just the module product. (Contributed by Stefan O'Rear, 1-Feb-2015.)
m s

Theoremdsmmelbas 27073* 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 27074 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 27075 The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s       m

Theoremprdsinvgd2 27076 Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
s

Theoremdsmmsubg 27077 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

Theoremdsmmlss 27078* The finite hull of a product of modules is additionally closed under scalar multiplication and thus is a linear subspace of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
Scalar        s              m

Theoremdsmmlmod 27079* The direct sum of a family of modules is a module. (Contributed by Stefan O'Rear, 11-Jan-2015.)
Scalar        m

19.16.44  Free modules

Syntaxcfrlm 27080 Class of free module generator.
freeLMod

Syntaxcuvc 27081 Class of basic unit vectors for an explicit free module.
unitVec

Definitiondf-frlm 27082* The -dimensional free module over a ring is the product of -many copies of the ring with componentwise addition and multiplication. If is infinite, the allowed vectors are restricted to those with finitely many nonzero coordinates; this ensures that the resulting module is actually spanned by its unit vectors. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod m ringLMod

Definitiondf-uvc 27083* unitVec is the unit vector in freeLMod along the axis. (Contributed by Stefan O'Rear, 1-Feb-2015.)
unitVec

Theoremfrlmval 27084 Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        m ringLMod

Theoremfrlmlmod 27085 The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmpws 27086 The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod               ringLMod s s

Theoremfrlmlss 27087 The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod               ringLMod s

Theoremfrlmpwsfi 27088 The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        ringLMod s

Theoremfrlmsca 27089 The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod        Scalar

Theoremfrlm0 27090 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 27087). (Contributed by Stefan O'Rear, 4-Feb-2015.)
freeLMod

Theoremfrlmbas 27091* Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmelbas 27092 Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod

Theoremfrlmrcl 27093 If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmbassup 27094 Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmbasmap 27095 Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmbasf 27096 Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmplusgval 27097 Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
freeLMod

Theoremfrlmvscafval 27098 Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
freeLMod

Theoremfrlmvscaval 27099 Scalar multiplication in a free module at a coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod

Theoremfrlmgsum 27100* Finite commutative sums in a free module are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 5-Jul-2015.)
freeLMod                                                         g g

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