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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | harinf 37601 | The Hartogs number of an infinite set is at least . MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) |
har | ||
Theorem | wdom2d2 37602* | Deduction for weak dominance by a Cartesian product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) |
* | ||
Theorem | ttac 37603 | Tarski's theorem about choice: infxpidm 9384 is equivalent to ax-ac 9281. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.) |
CHOICE | ||
Theorem | pw2f1ocnv 37604* | Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8067, 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.) |
Theorem | pw2f1o2 37605* | Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8067, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
Theorem | pw2f1o2val 37606* | Function value of the pw2f1o2 37605 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
Theorem | pw2f1o2val2 37607* | Membership in a mapped set under the pw2f1o2 37605 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
Theorem | soeq12d 37608 | Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
Theorem | freq12d 37609 | Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
Theorem | weeq12d 37610 | Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
Theorem | limsuc2 37611 | Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
Theorem | wepwsolem 37612* | Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
Theorem | wepwso 37613* | 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.) |
Theorem | dnnumch1 37614* | Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 8853. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | dnnumch2 37615* | Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | dnnumch3lem 37616* | Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | dnnumch3 37617* | Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | dnwech 37618* | Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | fnwe2val 37619* | Lemma for fnwe2 37623. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
Theorem | fnwe2lem1 37620* | Lemma for fnwe2 37623. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
Theorem | fnwe2lem2 37621* | Lemma for fnwe2 37623. 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.) |
Theorem | fnwe2lem3 37622* | Lemma for fnwe2 37623. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
Theorem | fnwe2 37623* | 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 7293 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
Theorem | aomclem1 37624* |
Lemma for dfac11 37632. 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.) |
Theorem | aomclem2 37625* | Lemma for dfac11 37632. Successor case 2, a choice function for subsets of . (Contributed by Stefan O'Rear, 18-Jan-2015.) |
Theorem | aomclem3 37626* | Lemma for dfac11 37632. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
recs | ||
Theorem | aomclem4 37627* | Lemma for dfac11 37632. Limit case. Patch together well-orderings constructed so far using fnwe2 37623 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
Theorem | aomclem5 37628* | Lemma for dfac11 37632. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
recs | ||
Theorem | aomclem6 37629* | Lemma for dfac11 37632. Transfinite induction, close over . (Contributed by Stefan O'Rear, 20-Jan-2015.) |
recs recs | ||
Theorem | aomclem7 37630* | Lemma for dfac11 37632. is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
recs recs | ||
Theorem | aomclem8 37631* | Lemma for dfac11 37632. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
Theorem | dfac11 37632* |
The right-hand side of this theorem (compare with ac4 9297),
sometimes
known as the "axiom of multiple choice", is a choice
equivalent.
Curiously, this statement cannot be proved without ax-reg 8497, 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 | ||
Theorem | kelac1 37633* | 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.) |
Theorem | kelac2lem 37634 | Lemma for kelac2 37635 and dfac21 37636: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Theorem | kelac2 37635* | Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Theorem | dfac21 37636 | 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 | ||
Syntax | clfig 37637 | Extend class notation with the class of finitely generated left modules. |
LFinGen | ||
Definition | df-lfig 37638 | 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 | ||
Theorem | islmodfg 37639* | Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
LFinGen | ||
Theorem | islssfg 37640* | Property of a finitely generated left (sub-)module. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
↾s LFinGen | ||
Theorem | islssfg2 37641* | 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 | ||
Theorem | islssfgi 37642 | Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
↾s LFinGen | ||
Theorem | fglmod 37643 | Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
LFinGen | ||
Theorem | lsmfgcl 37644 | The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
↾s ↾s ↾s LFinGen LFinGen LFinGen | ||
Syntax | clnm 37645 | Extend class notation with the class of Noetherian left modules. |
LNoeM | ||
Definition | df-lnm 37646* | A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
LNoeM ↾s LFinGen | ||
Theorem | islnm 37647* | Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
LNoeM ↾s LFinGen | ||
Theorem | islnm2 37648* | Property of being a Noetherian left module with finite generation expanded in terms of spans. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
LNoeM | ||
Theorem | lnmlmod 37649 | A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
LNoeM | ||
Theorem | lnmlssfg 37650 | A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
↾s LNoeM LFinGen | ||
Theorem | lnmlsslnm 37651 | All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
↾s LNoeM LNoeM | ||
Theorem | lnmfg 37652 | A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
LNoeM LFinGen | ||
Theorem | kercvrlsm 37653 | 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 | ||
Theorem | lmhmfgima 37654 | A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
↾s ↾s LFinGen LMHom LFinGen | ||
Theorem | lnmepi 37655 | Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
LMHom LNoeM LNoeM | ||
Theorem | lmhmfgsplit 37656 | 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 | ||
Theorem | lmhmlnmsplit 37657 | 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 | ||
Theorem | lnmlmic 37658 | Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
𝑚 LNoeM LNoeM | ||
Theorem | pwssplit4 37659* | Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
s s s ↾s LMIso | ||
Theorem | filnm 37660 | Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
LNoeM | ||
Theorem | pwslnmlem0 37661 | Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
s LNoeM | ||
Theorem | pwslnmlem1 37662* | First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
s LNoeM LNoeM | ||
Theorem | pwslnmlem2 37663 | A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
s s s LNoeM LNoeM LNoeM | ||
Theorem | pwslnm 37664 | Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
s LNoeM LNoeM | ||
Theorem | unxpwdom3 37665* | Weaker version of unxpwdom 8494 where a function is required only to be cancellative, not an injection. and are to be thought of as "large" "horizonal" sets, the others as "small". Because the operator is row-wise injective, but the whole row cannot inject into , each row must hit an element of ; by column injectivity, each row can be identified in at least one way by the element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE |
* | ||
Theorem | pwfi2f1o 37666* | The pw2f1o 8065 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.) |
finSupp | ||
Theorem | pwfi2en 37667* | Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.) |
finSupp | ||
Theorem | frlmpwfi 37668 | Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Proof shortened by AV, 14-Jun-2020.) |
ℤ/nℤ freeLMod | ||
Theorem | gicabl 37669 | Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) |
𝑔 | ||
Theorem | imasgim 37670 | A relabeling of the elements of a group induces an isomorphism to the relabeled group. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) (Revised by Mario Carneiro, 11-Aug-2015.) |
s GrpIso | ||
Theorem | isnumbasgrplem1 37671 | A set which is equipollent to the base set of a definable Abelian group is the base set of some (relabeled) Abelian group. (Contributed by Stefan O'Rear, 8-Jul-2015.) |
Theorem | harn0 37672 | The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.) |
har | ||
Theorem | numinfctb 37673 | A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.) |
Theorem | isnumbasgrplem2 37674 | If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
har | ||
Theorem | isnumbasgrplem3 37675 | Every nonempty numerable set can be given the structure of an Abelian group, either a finite cyclic group or a vector space over Z/2Z. (Contributed by Stefan O'Rear, 10-Jul-2015.) |
Theorem | isnumbasabl 37676 | A set is numerable iff it and its Hartogs number can be jointly given the structure of an Abelian group. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
har | ||
Theorem | isnumbasgrp 37677 | A set is numerable iff it and its Hartogs number can be jointly given the structure of a group. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
har | ||
Theorem | dfacbasgrp 37678 | A choice equivalent in abstract algebra: All nonempty sets admit a group structure. From http://mathoverflow.net/a/12988. (Contributed by Stefan O'Rear, 9-Jul-2015.) |
CHOICE | ||
Syntax | clnr 37679 | Extend class notation with the class of left Noetherian rings. |
LNoeR | ||
Definition | df-lnr 37680 | A ring is left-Noetherian iff it is Noetherian as a left module over itself. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
LNoeR ringLMod LNoeM | ||
Theorem | islnr 37681 | Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
LNoeR ringLMod LNoeM | ||
Theorem | lnrring 37682 | Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
LNoeR | ||
Theorem | lnrlnm 37683 | Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
LNoeR ringLMod LNoeM | ||
Theorem | islnr2 37684* | Property of being a left-Noetherian ring in terms of finite generation of ideals (the usual "pure ring theory" definition). (Contributed by Stefan O'Rear, 24-Jan-2015.) |
LIdeal RSpan LNoeR | ||
Theorem | islnr3 37685 | Relate left-Noetherian rings to Noetherian-type closure property of the left ideal system. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
LIdeal LNoeR NoeACS | ||
Theorem | lnr2i 37686* | Given an ideal in a left-Noetherian ring, there is a finite subset which generates it. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
LIdeal RSpan LNoeR | ||
Theorem | lpirlnr 37687 | Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
LPIR LNoeR | ||
Theorem | lnrfrlm 37688 | Finite-dimensional free modules over a Noetherian ring are Noetherian. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
freeLMod LNoeR LNoeM | ||
Theorem | lnrfg 37689 | Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.) |
Scalar LFinGen LNoeR LNoeM | ||
Theorem | lnrfgtr 37690 | A submodule of a finitely generated module over a Noetherian ring is finitely generated. Often taken as the definition of Noetherian ring. (Contributed by Stefan O'Rear, 7-Feb-2015.) |
Scalar ↾s LFinGen LNoeR LFinGen | ||
Syntax | cldgis 37691 | The leading ideal sequence used in the Hilbert Basis Theorem. |
ldgIdlSeq | ||
Definition | df-ldgis 37692* | Define a function which carries polynomial ideals to the sequence of coefficient ideals of leading coefficients of degree- elements in the polynomial ideal. The proof that this map is strictly monotone is the core of the Hilbert Basis Theorem hbt 37700. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
ldgIdlSeq LIdealPoly1 deg1 coe1 | ||
Theorem | hbtlem1 37693* | Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Poly1 LIdeal ldgIdlSeq deg1 coe1 | ||
Theorem | hbtlem2 37694 | Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
Poly1 LIdeal ldgIdlSeq LIdeal | ||
Theorem | hbtlem7 37695 | Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
Poly1 LIdeal ldgIdlSeq LIdeal | ||
Theorem | hbtlem4 37696 | The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
Poly1 LIdeal ldgIdlSeq | ||
Theorem | hbtlem3 37697 | The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Poly1 LIdeal ldgIdlSeq | ||
Theorem | hbtlem5 37698* | The leading ideal function is strictly monotone. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
Poly1 LIdeal ldgIdlSeq | ||
Theorem | hbtlem6 37699* | There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
Poly1 LIdeal ldgIdlSeq RSpan LNoeR | ||
Theorem | hbt 37700 | The Hilbert Basis Theorem - the ring of univariate polynomials over a Noetherian ring is a Noetherian ring. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
Poly1 LNoeR LNoeR |
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