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Theorem List for Metamath Proof Explorer - 35601-35700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremttac 35601 Tarski's theorem about choice: infxpidm 8976 is equivalent to ax-ac 8878. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
 |-  (CHOICE  <->  A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c ) )
 
Theorempw2f1ocnv 35602* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7676, 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.)
 |-  F  =  ( x  e.  ( 2o  ^m  A )  |->  ( `' x " { 1o } ) )   =>    |-  ( A  e.  V  ->  ( F : ( 2o  ^m  A ) -1-1-onto-> ~P A  /\  `' F  =  ( y  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  y ,  1o ,  (/) ) ) ) ) )
 
Theorempw2f1o2 35603* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7676, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  =  ( x  e.  ( 2o  ^m  A )  |->  ( `' x " { 1o } ) )   =>    |-  ( A  e.  V  ->  F : ( 2o 
 ^m  A ) -1-1-onto-> ~P A )
 
Theorempw2f1o2val 35604* Function value of the pw2f1o2 35603 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  F  =  ( x  e.  ( 2o  ^m  A )  |->  ( `' x " { 1o } ) )   =>    |-  ( X  e.  ( 2o  ^m  A )  ->  ( F `  X )  =  ( `' X " { 1o } )
 )
 
Theorempw2f1o2val2 35605* Membership in a mapped set under the pw2f1o2 35603 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  F  =  ( x  e.  ( 2o  ^m  A )  |->  ( `' x " { 1o } ) )   =>    |-  ( ( X  e.  ( 2o  ^m  A ) 
 /\  Y  e.  A )  ->  ( Y  e.  ( F `  X )  <-> 
 ( X `  Y )  =  1o )
 )
 
Theoremsoeq12d 35606 Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  ( ph  ->  R  =  S )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( R  Or  A  <->  S  Or  B ) )
 
Theoremfreq12d 35607 Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  ( ph  ->  R  =  S )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( R  Fr  A  <->  S  Fr  B ) )
 
Theoremweeq12d 35608 Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  ( ph  ->  R  =  S )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( R  We  A  <->  S  We  B ) )
 
Theoremlimsuc2 35609 Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  (
 ( Ord  A  /\  A  =  U. A ) 
 ->  ( B  e.  A  <->  suc 
 B  e.  A ) )
 
Theoremwepwsolem 35610* Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  ( ( z  e.  y  /\  -.  z  e.  x )  /\  A. w  e.  A  ( w R z  ->  ( w  e.  x  <->  w  e.  y ) ) ) }   &    |-  U  =  { <. x ,  y >.  | 
 E. z  e.  A  ( ( x `  z )  _E  (
 y `  z )  /\  A. w  e.  A  ( w R z  ->  ( x `  w )  =  ( y `  w ) ) ) }   &    |-  F  =  ( a  e.  ( 2o 
 ^m  A )  |->  ( `' a " { 1o } ) )   =>    |-  ( A  e.  _V  ->  F  Isom  U ,  T  ( ( 2o  ^m  A ) ,  ~P A ) )
 
Theoremwepwso 35611* A well-ordering induces a strict ordering on the power set. EDITORIAL: when well-orderings are set like, this can be strengthened to remove  A  e.  V. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  T  =  { <. x ,  y >.  |  E. z  e.  A  ( ( z  e.  y  /\  -.  z  e.  x )  /\  A. w  e.  A  ( w R z  ->  ( w  e.  x  <->  w  e.  y ) ) ) }   =>    |-  ( ( A  e.  V  /\  R  We  A )  ->  T  Or  ~P A )
 
Theoremdnnumch1 35612* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 8450 (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  = recs ( ( z  e. 
 _V  |->  ( G `  ( A  \  ran  z
 ) ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )   =>    |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
 
Theoremdnnumch2 35613* Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  = recs ( ( z  e. 
 _V  |->  ( G `  ( A  \  ran  z
 ) ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )   =>    |-  ( ph  ->  A 
 C_  ran  F )
 
Theoremdnnumch3lem 35614* Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  = recs ( ( z  e. 
 _V  |->  ( G `  ( A  \  ran  z
 ) ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )   =>    |-  ( ( ph  /\  w  e.  A ) 
 ->  ( ( x  e.  A  |->  |^| ( `' F " { x } )
 ) `  w )  =  |^| ( `' F " { w } )
 )
 
Theoremdnnumch3 35615* Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  = recs ( ( z  e. 
 _V  |->  ( G `  ( A  \  ran  z
 ) ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A -1-1-> On )
 
Theoremdnwech 35616* Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  F  = recs ( ( z  e. 
 _V  |->  ( G `  ( A  \  ran  z
 ) ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )   &    |-  H  =  { <. v ,  w >.  |  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) }   =>    |-  ( ph  ->  H  We  A )
 
Theoremfnwe2val 35617* Lemma for fnwe2 35621. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  (
 z  =  ( F `
  x )  ->  S  =  U )   &    |-  T  =  { <. x ,  y >.  |  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x U y ) ) }   =>    |-  (
 a T b  <->  ( ( F `
  a ) R ( F `  b
 )  \/  ( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a ) 
 /  z ]_ S b ) ) )
 
Theoremfnwe2lem1 35618* Lemma for fnwe2 35621. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  (
 z  =  ( F `
  x )  ->  S  =  U )   &    |-  T  =  { <. x ,  y >.  |  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x U y ) ) }   &    |-  (
 ( ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y
 )  =  ( F `
  x ) }
 )   =>    |-  ( ( ph  /\  a  e.  A )  ->  [_ ( F `  a )  /  z ]_ S  We  {
 y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
 
Theoremfnwe2lem2 35619* Lemma for fnwe2 35621. An element which is in a minimal fiber and minimal within its fiber is minimal globally; thus  T is well-founded. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  (
 z  =  ( F `
  x )  ->  S  =  U )   &    |-  T  =  { <. x ,  y >.  |  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x U y ) ) }   &    |-  (
 ( ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y
 )  =  ( F `
  x ) }
 )   &    |-  ( ph  ->  ( F  |`  A ) : A --> B )   &    |-  ( ph  ->  R  We  B )   &    |-  ( ph  ->  a  C_  A )   &    |-  ( ph  ->  a  =/=  (/) )   =>    |-  ( ph  ->  E. b  e.  a  A. c  e.  a  -.  c T b )
 
Theoremfnwe2lem3 35620* Lemma for fnwe2 35621. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  (
 z  =  ( F `
  x )  ->  S  =  U )   &    |-  T  =  { <. x ,  y >.  |  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x U y ) ) }   &    |-  (
 ( ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y
 )  =  ( F `
  x ) }
 )   &    |-  ( ph  ->  ( F  |`  A ) : A --> B )   &    |-  ( ph  ->  R  We  B )   &    |-  ( ph  ->  a  e.  A )   &    |-  ( ph  ->  b  e.  A )   =>    |-  ( ph  ->  ( a T b  \/  a  =  b  \/  b T a ) )
 
Theoremfnwe2 35621* 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 6914 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  (
 z  =  ( F `
  x )  ->  S  =  U )   &    |-  T  =  { <. x ,  y >.  |  ( ( F `
  x ) R ( F `  y
 )  \/  ( ( F `  x )  =  ( F `  y )  /\  x U y ) ) }   &    |-  (
 ( ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y
 )  =  ( F `
  x ) }
 )   &    |-  ( ph  ->  ( F  |`  A ) : A --> B )   &    |-  ( ph  ->  R  We  B )   =>    |-  ( ph  ->  T  We  A )
 
Theoremaomclem1 35622* Lemma for dfac11 35630. 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  ( R1 `  A ). In what follows,  A is the index of the rank we wish to well-order,  z is the collection of well-orderings constructed so far,  dom  z is the set of ordinal indexes of constructed ranks i.e. the next rank to construct, and  y 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.)

 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  ( ph  ->  dom  z  e.  On )   &    |-  ( ph  ->  dom  z  =  suc  U. dom  z )   &    |-  ( ph  ->  A. a  e. 
 dom  z ( z `
  a )  We  ( R1 `  a
 ) )   =>    |-  ( ph  ->  B  Or  ( R1 `  dom  z ) )
 
Theoremaomclem2 35623* Lemma for dfac11 35630. Successor case 2, a choice function for subsets of  ( R1 `  dom  z ). (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z ) ,  B ) )   &    |-  ( ph  ->  dom  z  e. 
 On )   &    |-  ( ph  ->  dom  z  =  suc  U. dom  z )   &    |-  ( ph  ->  A. a  e.  dom  z
 ( z `  a
 )  We  ( R1
 `  a ) )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  dom  z  C_  A )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
 y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  A. a  e. 
 ~P  ( R1 `  dom  z ) ( a  =/=  (/)  ->  ( C `  a )  e.  a
 ) )
 
Theoremaomclem3 35624* Lemma for dfac11 35630. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z ) ,  B ) )   &    |-  D  = recs ( (
 a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
 )  \  ran  a ) ) ) )   &    |-  E  =  { <. a ,  b >.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }   &    |-  ( ph  ->  dom  z  e.  On )   &    |-  ( ph  ->  dom  z  =  suc  U. dom  z )   &    |-  ( ph  ->  A. a  e.  dom  z
 ( z `  a
 )  We  ( R1
 `  a ) )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  dom  z  C_  A )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
 y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  E  We  ( R1 `  dom  z
 ) )
 
Theoremaomclem4 35625* Lemma for dfac11 35630. Limit case. Patch together well-orderings constructed so far using fnwe2 35621 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  F  =  { <. a ,  b >.  |  ( ( rank `  a )  _E  ( rank `  b )  \/  ( ( rank `  a
 )  =  ( rank `  b )  /\  a
 ( z `  suc  ( rank `  a )
 ) b ) ) }   &    |-  ( ph  ->  dom  z  e.  On )   &    |-  ( ph  ->  dom  z  =  U.
 dom  z )   &    |-  ( ph  ->  A. a  e.  dom  z ( z `  a )  We  ( R1 `  a ) )   =>    |-  ( ph  ->  F  We  ( R1 `  dom  z
 ) )
 
Theoremaomclem5 35626* Lemma for dfac11 35630. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z ) ,  B ) )   &    |-  D  = recs ( (
 a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
 )  \  ran  a ) ) ) )   &    |-  E  =  { <. a ,  b >.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }   &    |-  F  =  { <. a ,  b >.  |  ( ( rank `  a )  _E  ( rank `  b )  \/  ( ( rank `  a
 )  =  ( rank `  b )  /\  a
 ( z `  suc  ( rank `  a )
 ) b ) ) }   &    |-  G  =  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  ( ( R1
 `  dom  z )  X.  ( R1 `  dom  z ) ) )   &    |-  ( ph  ->  dom  z  e. 
 On )   &    |-  ( ph  ->  A. a  e.  dom  z
 ( z `  a
 )  We  ( R1
 `  a ) )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  dom  z  C_  A )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
 y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  G  We  ( R1 `  dom  z
 ) )
 
Theoremaomclem6 35627* Lemma for dfac11 35630. Transfinite induction, close over  z. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z ) ,  B ) )   &    |-  D  = recs ( (
 a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
 )  \  ran  a ) ) ) )   &    |-  E  =  { <. a ,  b >.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }   &    |-  F  =  { <. a ,  b >.  |  ( ( rank `  a )  _E  ( rank `  b )  \/  ( ( rank `  a
 )  =  ( rank `  b )  /\  a
 ( z `  suc  ( rank `  a )
 ) b ) ) }   &    |-  G  =  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  ( ( R1
 `  dom  z )  X.  ( R1 `  dom  z ) ) )   &    |-  H  = recs ( (
 z  e.  _V  |->  G ) )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
 y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  ( H `  A )  We  ( R1 `  A ) )
 
Theoremaomclem7 35628* Lemma for dfac11 35630. 
( R1 `  A
) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  B  =  { <. a ,  b >.  |  E. c  e.  ( R1 `  U. dom  z ) ( ( c  e.  b  /\  -.  c  e.  a ) 
 /\  A. d  e.  ( R1 `  U. dom  z
 ) ( d ( z `  U. dom  z ) c  ->  ( d  e.  a  <->  d  e.  b ) ) ) }   &    |-  C  =  ( a  e.  _V  |->  sup ( ( y `  a ) ,  ( R1 `  dom  z ) ,  B ) )   &    |-  D  = recs ( (
 a  e.  _V  |->  ( C `  ( ( R1 `  dom  z
 )  \  ran  a ) ) ) )   &    |-  E  =  { <. a ,  b >.  |  |^| ( `' D " { a } )  e.  |^| ( `' D " { b } ) }   &    |-  F  =  { <. a ,  b >.  |  ( ( rank `  a )  _E  ( rank `  b )  \/  ( ( rank `  a
 )  =  ( rank `  b )  /\  a
 ( z `  suc  ( rank `  a )
 ) b ) ) }   &    |-  G  =  ( if ( dom  z  =  U. dom  z ,  F ,  E )  i^i  ( ( R1
 `  dom  z )  X.  ( R1 `  dom  z ) ) )   &    |-  H  = recs ( (
 z  e.  _V  |->  G ) )   &    |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  (
 y `  a )  e.  ( ( ~P a  i^i  Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  E. b  b  We  ( R1 `  A ) )
 
Theoremaomclem8 35629* Lemma for dfac11 35630. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. a  e.  ~P  ( R1 `  A ) ( a  =/=  (/)  ->  ( y `  a )  e.  (
 ( ~P a  i^i 
 Fin )  \  { (/)
 } ) ) )   =>    |-  ( ph  ->  E. b  b  We  ( R1 `  A ) )
 
Theoremdfac11 35630* The right-hand side of this theorem (compare with ac4 8894), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8098, 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  <->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  (
 f `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
 } ) ) )
 
Theoremkelac1 35631* 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.)
 |-  (
 ( ph  /\  x  e.  I )  ->  S  =/= 
 (/) )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  J  e.  Top )   &    |-  (
 ( ph  /\  x  e.  I )  ->  C  e.  ( Clsd `  J )
 )   &    |-  ( ( ph  /\  x  e.  I )  ->  B : S -1-1-onto-> C )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  U  e.  U. J )   &    |-  ( ph  ->  ( Xt_ `  ( x  e.  I  |->  J ) )  e.  Comp )   =>    |-  ( ph  ->  X_ x  e.  I  S  =/=  (/) )
 
Theoremkelac2lem 35632 Lemma for kelac2 35633 and dfac21 35634: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( S  e.  V  ->  (
 topGen `  { S ,  { ~P U. S } } )  e.  Comp )
 
Theoremkelac2 35633* Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  (
 ( ph  /\  x  e.  I )  ->  S  e.  V )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  S  =/=  (/) )   &    |-  ( ph  ->  ( Xt_ `  ( x  e.  I  |->  (
 topGen `  { S ,  { ~P U. S } } ) ) )  e.  Comp )   =>    |-  ( ph  ->  X_ x  e.  I  S  =/=  (/) )
 
Theoremdfac21 35634 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  <->  A. f ( f : dom  f --> Comp  ->  (
 Xt_ `  f )  e.  Comp ) )
 
21.23.38  Finitely generated left modules
 
Syntaxclfig 35635 Extend class notation with the class of finitely generated left modules.
 class LFinGen
 
Definitiondf-lfig 35636 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  =  { w  e.  LMod  |  (
 Base `  w )  e.  ( ( LSpan `  w ) " ( ~P ( Base `  w )  i^i 
 Fin ) ) }
 
Theoremislmodfg 35637* Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  B  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( W  e.  LMod  ->  ( W  e. LFinGen  <->  E. b  e.  ~P  B ( b  e. 
 Fin  /\  ( N `  b )  =  B ) ) )
 
Theoremislssfg 35638* Property of a finitely generated left (sub-)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  S ) 
 ->  ( X  e. LFinGen  <->  E. b  e.  ~P  U ( b  e. 
 Fin  /\  ( N `  b )  =  U ) ) )
 
Theoremislssfg2 35639* Property of a finitely generated left (sub-)module, with a relaxed constraint on the spanning vectors. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  B  =  ( Base `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  S ) 
 ->  ( X  e. LFinGen  <->  E. b  e.  ( ~P B  i^i  Fin )
 ( N `  b
 )  =  U ) )
 
Theoremislssfgi 35640 Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  N  =  ( LSpan `  W )   &    |-  V  =  ( Base `  W )   &    |-  X  =  ( Ws  ( N `  B ) )   =>    |-  ( ( W  e.  LMod  /\  B  C_  V  /\  B  e.  Fin )  ->  X  e. LFinGen )
 
Theoremfglmod 35641 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( M  e. LFinGen  ->  M  e.  LMod
 )
 
Theoremlsmfgcl 35642 The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  U  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  D  =  ( Ws  A )   &    |-  E  =  ( Ws  B )   &    |-  F  =  ( Ws  ( A  .(+)  B ) )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   &    |-  ( ph  ->  D  e. LFinGen )   &    |-  ( ph  ->  E  e. LFinGen )   =>    |-  ( ph  ->  F  e. LFinGen )
 
21.23.39  Noetherian left modules I
 
Syntaxclnm 35643 Extend class notation with the class of Noetherian left modules.
 class LNoeM
 
Definitiondf-lnm 35644* A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |- LNoeM  =  { w  e.  LMod  |  A. i  e.  ( LSubSp `  w ) ( ws  i )  e. LFinGen }
 
Theoremislnm 35645* Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  S  =  ( LSubSp `  M )   =>    |-  ( M  e. LNoeM  <->  ( M  e.  LMod  /\  A. i  e.  S  ( Ms  i )  e. LFinGen )
 )
 
Theoremislnm2 35646* Property of being a Noetherian left module with finite generation expanded in terms of spans. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  B  =  ( Base `  M )   &    |-  S  =  ( LSubSp `  M )   &    |-  N  =  ( LSpan `  M )   =>    |-  ( M  e. LNoeM  <->  ( M  e.  LMod  /\  A. i  e.  S  E. g  e.  ( ~P B  i^i  Fin )
 i  =  ( N `
  g ) ) )
 
Theoremlnmlmod 35647 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  ( M  e. LNoeM  ->  M  e.  LMod
 )
 
Theoremlnmlssfg 35648 A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  S  =  ( LSubSp `  M )   &    |-  R  =  ( Ms  U )   =>    |-  ( ( M  e. LNoeM  /\  U  e.  S ) 
 ->  R  e. LFinGen )
 
Theoremlnmlsslnm 35649 All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  S  =  ( LSubSp `  M )   &    |-  R  =  ( Ms  U )   =>    |-  ( ( M  e. LNoeM  /\  U  e.  S ) 
 ->  R  e. LNoeM )
 
Theoremlnmfg 35650 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  ( M  e. LNoeM  ->  M  e. LFinGen )
 
Theoremkercvrlsm 35651 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.)
 |-  U  =  ( LSubSp `  S )   &    |-  .(+)  =  (
 LSSum `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  K  =  ( `' F " {  .0.  } )   &    |-  B  =  (
 Base `  S )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  D  e.  U )   &    |-  ( ph  ->  ( F " D )  = 
 ran  F )   =>    |-  ( ph  ->  ( K  .(+)  D )  =  B )
 
Theoremlmhmfgima 35652 A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( Ts  ( F " A ) )   &    |-  X  =  ( Ss  A )   &    |-  U  =  (
 LSubSp `  S )   &    |-  ( ph  ->  X  e. LFinGen )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   =>    |-  ( ph  ->  Y  e. LFinGen )
 
Theoremlnmepi 35653 Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  B  =  ( Base `  T )   =>    |-  (
 ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  T  e. LNoeM )
 
Theoremlmhmfgsplit 35654 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.)
 |-  .0.  =  ( 0g `  T )   &    |-  K  =  ( `' F " {  .0.  } )   &    |-  U  =  ( Ss  K )   &    |-  V  =  ( Ts 
 ran  F )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen ) 
 ->  S  e. LFinGen )
 
Theoremlmhmlnmsplit 35655 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.)
 |-  .0.  =  ( 0g `  T )   &    |-  K  =  ( `' F " {  .0.  } )   &    |-  U  =  ( Ss  K )   &    |-  V  =  ( Ts 
 ran  F )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM ) 
 ->  S  e. LNoeM )
 
Theoremlnmlmic 35656 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑚 
 S  ->  ( R  e. LNoeM  <->  S  e. LNoeM ) )
 
21.23.40  Addenda for structure powers
 
Theorempwssplit4 35657* Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  E  =  ( R  ^s  ( A  u.  B ) )   &    |-  G  =  ( Base `  E )   &    |-  .0.  =  ( 0g `  R )   &    |-  K  =  { y  e.  G  |  ( y  |`  A )  =  ( A  X.  {  .0.  } ) }   &    |-  F  =  ( x  e.  K  |->  ( x  |`  B )
 )   &    |-  C  =  ( R 
 ^s 
 A )   &    |-  D  =  ( R  ^s  B )   &    |-  L  =  ( Es  K )   =>    |-  ( ( R  e.  LMod  /\  ( A  u.  B )  e.  V  /\  ( A  i^i  B )  =  (/) )  ->  F  e.  ( L LMIso  D ) )
 
Theoremfilnm 35658 Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  B  =  ( Base `  W )   =>    |-  (
 ( W  e.  LMod  /\  B  e.  Fin )  ->  W  e. LNoeM )
 
Theorempwslnmlem0 35659 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  (/) )   =>    |-  ( W  e.  LMod 
 ->  Y  e. LNoeM )
 
Theorempwslnmlem1 35660* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  { i } )   =>    |-  ( W  e. LNoeM  ->  Y  e. LNoeM )
 
Theorempwslnmlem2 35661 A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  X  =  ( W  ^s  A )   &    |-  Y  =  ( W  ^s  B )   &    |-  Z  =  ( W  ^s  ( A  u.  B ) )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  X  e. LNoeM )   &    |-  ( ph  ->  Y  e. LNoeM )   =>    |-  ( ph  ->  Z  e. LNoeM )
 
Theorempwslnm 35662 Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  I )   =>    |-  (
 ( W  e. LNoeM  /\  I  e.  Fin )  ->  Y  e. LNoeM )
 
21.23.41  Every set admits a group structure iff choice
 
Theoremunxpwdom3 35663* Weaker version of unxpwdom 8095 where a function is required only to be cancellative, not an injection.  D and  B 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  A, each row must hit an element of 
B; by column injectivity, each row can be identified in at least one way by the  B element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  D  e.  X )   &    |-  (
 ( ph  /\  a  e.  C  /\  b  e.  D )  ->  (
 a  .+  b )  e.  ( A  u.  B ) )   &    |-  ( ( (
 ph  /\  a  e.  C )  /\  ( b  e.  D  /\  c  e.  D ) )  ->  ( ( a  .+  b )  =  (
 a  .+  c )  <->  b  =  c ) )   &    |-  ( ( ( ph  /\  d  e.  D ) 
 /\  ( a  e.  C  /\  c  e.  C ) )  ->  ( ( c  .+  d )  =  (
 a  .+  d )  <->  c  =  a ) )   &    |-  ( ph  ->  -.  D  ~<_  A )   =>    |-  ( ph  ->  C  ~<_*  ( D  X.  B ) )
 
Theorempwfi2f1o 35664* The pw2f1o 7674 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.)
 |-  S  =  { y  e.  ( 2o  ^m  A )  |  y finSupp  (/) }   &    |-  F  =  ( x  e.  S  |->  ( `' x " { 1o } ) )   =>    |-  ( A  e.  V  ->  F : S -1-1-onto-> ( ~P A  i^i  Fin ) )
 
Theorempwfi2en 35665* Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.)
 |-  S  =  { y  e.  ( 2o  ^m  A )  |  y finSupp  (/) }   =>    |-  ( A  e.  V  ->  S  ~~  ( ~P A  i^i  Fin )
 )
 
Theoremfrlmpwfi 35666 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.)
 |-  R  =  (ℤ/n `  2 )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  Y )   =>    |-  ( I  e.  V  ->  B  ~~  ( ~P I  i^i  Fin )
 )
 
Theoremgicabl 35667 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  ( G  ~=g𝑔 
 H  ->  ( G  e.  Abel 
 <->  H  e.  Abel )
 )
 
Theoremimasgim 35668 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.)
 |-  ( ph  ->  U  =  ( F  "s  R ) )   &    |-  ( ph  ->  V  =  (
 Base `  R ) )   &    |-  ( ph  ->  F : V
 -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  F  e.  ( R GrpIso  U ) )
 
Theorembasfn 35669 Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  Base  Fn 
 _V
 
Theoremisnumbasgrplem1 35670 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.)
 |-  B  =  ( Base `  R )   =>    |-  (
 ( R  e.  Abel  /\  C  ~~  B ) 
 ->  C  e.  ( Base "
 Abel ) )
 
Theoremharn0 35671 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( S  e.  V  ->  (har `  S )  =/=  (/) )
 
Theoremnuminfctb 35672 A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  (
 ( S  e.  dom  card  /\  -.  S  e.  Fin )  ->  om  ~<_  S )
 
Theoremisnumbasgrplem2 35673 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.)
 |-  (
 ( S  u.  (har `  S ) )  e.  ( Base " Grp )  ->  S  e.  dom  card )
 
Theoremisnumbasgrplem3 35674 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.)
 |-  (
 ( S  e.  dom  card  /\  S  =/=  (/) )  ->  S  e.  ( Base "
 Abel ) )
 
Theoremisnumbasabl 35675 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.)
 |-  ( S  e.  dom  card  <->  ( S  u.  (har `  S ) )  e.  ( Base " Abel ) )
 
Theoremisnumbasgrp 35676 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.)
 |-  ( S  e.  dom  card  <->  ( S  u.  (har `  S ) )  e.  ( Base " Grp ) )
 
Theoremdfacbasgrp 35677 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  <->  ( Base " Grp )  =  ( _V  \  { (/) } ) )
 
21.23.42  Noetherian rings and left modules II
 
Syntaxclnr 35678 Extend class notation with the class of left Noetherian rings.
 class LNoeR
 
Definitiondf-lnr 35679 A ring is left-Noetherian iff it is Noetherian as a left module over itself. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |- LNoeR  =  {
 a  e.  Ring  |  (ringLMod `  a )  e. LNoeM }
 
Theoremislnr 35680 Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  <->  ( A  e.  Ring  /\  (ringLMod `  A )  e. LNoeM ) )
 
Theoremlnrring 35681 Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  ->  A  e.  Ring
 )
 
Theoremlnrlnm 35682 Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( A  e. LNoeR  ->  (ringLMod `  A )  e. LNoeM )
 
Theoremislnr2 35683* 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.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (LIdeal `  R )   &    |-  N  =  (RSpan `  R )   =>    |-  ( R  e. LNoeR  <->  ( R  e.  Ring  /\  A. i  e.  U  E. g  e.  ( ~P B  i^i  Fin )
 i  =  ( N `
  g ) ) )
 
Theoremislnr3 35684 Relate left-Noetherian rings to Noetherian-type closure property of the left ideal system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e. LNoeR  <->  ( R  e.  Ring  /\  U  e.  (NoeACS `  B ) ) )
 
Theoremlnr2i 35685* Given an ideal in a left-Noetherian ring, there is a finite subset which generates it. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  N  =  (RSpan `  R )   =>    |-  (
 ( R  e. LNoeR  /\  I  e.  U )  ->  E. g  e.  ( ~P I  i^i  Fin ) I  =  ( N `  g ) )
 
Theoremlpirlnr 35686 Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( R  e. LPIR  ->  R  e. LNoeR )
 
Theoremlnrfrlm 35687 Finite-dimensional free modules over a Noetherian ring are Noetherian. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  Y  =  ( R freeLMod  I )   =>    |-  (
 ( R  e. LNoeR  /\  I  e.  Fin )  ->  Y  e. LNoeM )
 
Theoremlnrfg 35688 Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.)
 |-  S  =  (Scalar `  M )   =>    |-  (
 ( M  e. LFinGen  /\  S  e. LNoeR )  ->  M  e. LNoeM )
 
Theoremlnrfgtr 35689 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.)
 |-  S  =  (Scalar `  M )   &    |-  U  =  ( LSubSp `  M )   &    |-  N  =  ( Ms  P )   =>    |-  ( ( M  e. LFinGen  /\  S  e. LNoeR  /\  P  e.  U )  ->  N  e. LFinGen )
 
21.23.43  Hilbert's Basis Theorem
 
Syntaxcldgis 35690 The leading ideal sequence used in the Hilbert Basis Theorem.
 class ldgIdlSeq
 
Definitiondf-ldgis 35691* Define a function which carries polynomial ideals to the sequence of coefficient ideals of leading coefficients of degree-  x elements in the polynomial ideal. The proof that this map is strictly monotone is the core of the Hilbert Basis Theorem hbt 35699. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |- ldgIdlSeq  =  ( r  e.  _V  |->  ( i  e.  (LIdeal `  (Poly1 `  r ) )  |->  ( x  e.  NN0  |->  { j  |  E. k  e.  i  ( ( ( deg1  `  r
 ) `  k )  <_  x  /\  j  =  ( (coe1 `  k ) `  x ) ) }
 ) ) )
 
Theoremhbtlem1 35692* Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  D  =  ( deg1  `  R )   =>    |-  (
 ( R  e.  V  /\  I  e.  U  /\  X  e.  NN0 )  ->  ( ( S `  I ) `  X )  =  { j  |  E. k  e.  I  ( ( D `  k )  <_  X  /\  j  =  ( (coe1 `  k ) `  X ) ) } )
 
Theoremhbtlem2 35693 Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  T  =  (LIdeal `  R )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  U  /\  X  e.  NN0 )  ->  ( ( S `  I ) `  X )  e.  T )
 
Theoremhbtlem7 35694 Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  T  =  (LIdeal `  R )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  U ) 
 ->  ( S `  I
 ) : NN0 --> T )
 
Theoremhbtlem4 35695 The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  U )   &    |-  ( ph  ->  X  e.  NN0 )   &    |-  ( ph  ->  Y  e.  NN0 )   &    |-  ( ph  ->  X 
 <_  Y )   =>    |-  ( ph  ->  (
 ( S `  I
 ) `  X )  C_  ( ( S `  I ) `  Y ) )
 
Theoremhbtlem3 35696 The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  U )   &    |-  ( ph  ->  J  e.  U )   &    |-  ( ph  ->  I  C_  J )   &    |-  ( ph  ->  X  e.  NN0 )   =>    |-  ( ph  ->  (
 ( S `  I
 ) `  X )  C_  ( ( S `  J ) `  X ) )
 
Theoremhbtlem5 35697* The leading ideal function is strictly monotone. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  U )   &    |-  ( ph  ->  J  e.  U )   &    |-  ( ph  ->  I  C_  J )   &    |-  ( ph  ->  A. x  e.  NN0  ( ( S `
  J ) `  x )  C_  ( ( S `  I ) `
  x ) )   =>    |-  ( ph  ->  I  =  J )
 
Theoremhbtlem6 35698* There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  P  =  (Poly1 `  R )   &    |-  U  =  (LIdeal `  P )   &    |-  S  =  (ldgIdlSeq `  R )   &    |-  N  =  (RSpan `  P )   &    |-  ( ph  ->  R  e. LNoeR )   &    |-  ( ph  ->  I  e.  U )   &    |-  ( ph  ->  X  e.  NN0 )   =>    |-  ( ph  ->  E. k  e.  ( ~P I  i^i  Fin ) ( ( S `
  I ) `  X )  C_  ( ( S `  ( N `
  k ) ) `
  X ) )
 
Theoremhbt 35699 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.)
 |-  P  =  (Poly1 `  R )   =>    |-  ( R  e. LNoeR  ->  P  e. LNoeR )
 
21.23.44  Additional material on polynomials [DEPRECATED]
 
Syntaxcmnc 35700 Extend class notation with the class of monic polynomials.
 class  Monic
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