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Theorem List for Metamath Proof Explorer - 27101-27200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremjm2.27 27101* 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 27098 and jm2.27c 27100. Once Diophantine relations have been defined, the content of the theorem is "rmY is Diophantine" (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  =  ( A Yrm  B ) 
 <-> 
 E. d  e.  NN0  E. e  e.  NN0  E. f  e.  NN0  E. g  e. 
 NN0  E. h  e.  NN0  E. i  e.  NN0  E. j  e.  NN0  ( ( ( ( ( d ^
 2 )  -  (
 ( ( A ^
 2 )  -  1
 )  x.  ( C ^ 2 ) ) )  =  1  /\  ( ( f ^
 2 )  -  (
 ( ( A ^
 2 )  -  1
 )  x.  ( e ^ 2 ) ) )  =  1  /\  g  e.  ( ZZ>= `  2 ) )  /\  ( ( ( i ^ 2 )  -  ( ( ( g ^ 2 )  -  1 )  x.  ( h ^ 2 ) ) )  =  1  /\  e  =  ( (
 j  +  1 )  x.  ( 2  x.  ( C ^ 2
 ) ) )  /\  f  ||  ( g  -  A ) ) ) 
 /\  ( ( ( 2  x.  C ) 
 ||  ( g  -  1 )  /\  f  ||  ( h  -  C ) )  /\  ( ( 2  x.  C ) 
 ||  ( h  -  B )  /\  B  <_  C ) ) ) ) )
 
Theoremjm2.27dlem1 27102* Lemma for rmydioph 27107. Subsitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  ( 1 ... B )   =>    |-  ( a  =  ( b  |`  ( 1 ... B ) )  ->  ( a `  A )  =  ( b `  A ) )
 
Theoremjm2.27dlem2 27103 Lemma for rmydioph 27107. This theorem is used along with the next three to efficiently infer steps like 
7  e.  ( 1 ... 10 ). (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  ( 1 ... B )   &    |-  C  =  ( B  +  1 )   &    |-  B  e.  NN   =>    |-  A  e.  ( 1
 ... C )
 
Theoremjm2.27dlem3 27104 Lemma for rmydioph 27107. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  NN   =>    |-  A  e.  ( 1
 ... A )
 
Theoremjm2.27dlem4 27105 Lemma for rmydioph 27107. Infer  NN-hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  NN   &    |-  B  =  ( A  +  1 )   =>    |-  B  e.  NN
 
Theoremjm2.27dlem5 27106 Lemma for rmydioph 27107. Used with sselii 3177 to infer membership of midpoints of range; jm2.27dlem2 27103 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  B  =  ( A  +  1 )   &    |-  ( 1 ...
 B )  C_  (
 1 ... C )   =>    |-  ( 1 ...
 A )  C_  (
 1 ... C )
 
Theoremrmydioph 27107 jm2.27 27101 restated in terms of Diophantine sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  { a  e.  ( NN0  ^m  (
 1 ... 3 ) )  |  ( ( a `
  1 )  e.  ( ZZ>= `  2 )  /\  ( a `  3
 )  =  ( ( a `  1 ) Yrm  ( a `  2 ) ) ) }  e.  (Dioph `  3 )
 
18.17.37  X and Y sequences 5: Diophantine representability of X, ^, _C
 
Theoremrmxdiophlem 27108* X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN0  /\  X  e.  NN0 )  ->  ( X  =  ( A Xrm  N ) 
 <-> 
 E. y  e.  NN0  ( y  =  ( A Yrm 
 N )  /\  (
 ( X ^ 2
 )  -  ( ( ( A ^ 2
 )  -  1 )  x.  ( y ^
 2 ) ) )  =  1 ) ) )
 
Theoremrmxdioph 27109 X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  { a  e.  ( NN0  ^m  (
 1 ... 3 ) )  |  ( ( a `
  1 )  e.  ( ZZ>= `  2 )  /\  ( a `  3
 )  =  ( ( a `  1 ) Xrm  ( a `  2 ) ) ) }  e.  (Dioph `  3 )
 
Theoremjm3.1lem1 27110 Lemma for jm3.1 27113. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  ( ph  ->  A  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2 )
 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( K Yrm  ( N  +  1 ) )  <_  A )   =>    |-  ( ph  ->  ( K ^ N )  <  A )
 
Theoremjm3.1lem2 27111 Lemma for jm3.1 27113. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  ( ph  ->  A  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2 )
 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( K Yrm  ( N  +  1 ) )  <_  A )   =>    |-  ( ph  ->  ( K ^ N )  < 
 ( ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  -  1 ) )
 
Theoremjm3.1lem3 27112 Lemma for jm3.1 27113. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  ( ph  ->  A  e.  ( ZZ>=
 `  2 ) )   &    |-  ( ph  ->  K  e.  ( ZZ>= `  2 )
 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( K Yrm  ( N  +  1 ) )  <_  A )   =>    |-  ( ph  ->  (
 ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  -  1 )  e. 
 NN )
 
Theoremjm3.1 27113 Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 ( ( A  e.  ( ZZ>= `  2 )  /\  K  e.  ( ZZ>= `  2 )  /\  N  e.  NN )  /\  ( K Yrm  ( N  +  1 ) )  <_  A )  ->  ( K ^ N )  =  ( (
 ( A Xrm  N )  -  ( ( A  -  K )  x.  ( A Yrm 
 N ) ) ) 
 mod  ( ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  -  1 ) ) )
 
Theoremexpdiophlem1 27114* Lemma for expdioph 27116. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  ( C  e.  NN0  ->  (
 ( ( A  e.  ( ZZ>= `  2 )  /\  B  e.  NN )  /\  C  =  ( A ^ B ) )  <->  E. d  e.  NN0  E. e  e.  NN0  E. f  e. 
 NN0  ( ( A  e.  ( ZZ>= `  2
 )  /\  B  e.  NN )  /\  ( ( A  e.  ( ZZ>= `  2 )  /\  d  =  ( A Yrm  ( B  +  1 ) ) ) 
 /\  ( ( d  e.  ( ZZ>= `  2
 )  /\  e  =  ( d Yrm  B ) ) 
 /\  ( ( d  e.  ( ZZ>= `  2
 )  /\  f  =  ( d Xrm  B ) ) 
 /\  ( C  <  ( ( ( ( 2  x.  d )  x.  A )  -  ( A ^ 2 ) )  -  1 )  /\  ( ( ( ( 2  x.  d )  x.  A )  -  ( A ^ 2 ) )  -  1 ) 
 ||  ( ( f  -  ( ( d  -  A )  x.  e ) )  -  C ) ) ) ) ) ) ) )
 
Theoremexpdiophlem2 27115 Lemma for expdioph 27116. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  { a  e.  ( NN0  ^m  (
 1 ... 3 ) )  |  ( ( ( a `  1 )  e.  ( ZZ>= `  2
 )  /\  ( a `  2 )  e.  NN )  /\  ( a `  3 )  =  (
 ( a `  1
 ) ^ ( a `
  2 ) ) ) }  e.  (Dioph `  3 )
 
Theoremexpdioph 27116 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.)
 |-  { a  e.  ( NN0  ^m  (
 1 ... 3 ) )  |  ( a `  3 )  =  (
 ( a `  1
 ) ^ ( a `
  2 ) ) }  e.  (Dioph `  3 )
 
18.17.38  Uncategorized stuff not associated with a major project
 
Theoremsetindtr 27117* 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 7419; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( A. x ( x  C_  A  ->  x  e.  A )  ->  ( E. y
 ( Tr  y  /\  B  e.  y )  ->  B  e.  A ) )
 
Theoremsetindtrs 27118* Epsilon induction scheme without Infinity. See comments at setindtr 27117. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( A. y  e.  x  ps  ->  ph )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   &    |-  ( x  =  B  ->  ( ph  <->  ch ) )   =>    |-  ( E. z ( Tr  z  /\  B  e.  z )  ->  ch )
 
Theoremdford3lem1 27119* Lemma for dford3 27121. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  (
 ( Tr  N  /\  A. y  e.  N  Tr  y )  ->  A. b  e.  N  ( Tr  b  /\  A. y  e.  b  Tr  y ) )
 
Theoremdford3lem2 27120* Lemma for dford3 27121. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  (
 ( Tr  x  /\  A. y  e.  x  Tr  y )  ->  x  e. 
 On )
 
Theoremdford3 27121* Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( Ord  N  <->  ( Tr  N  /\  A. x  e.  N  Tr  x ) )
 
Theoremdford4 27122* dford3 27121 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( Ord  N  <->  A. a A. b A. c ( ( a  e.  N  /\  b  e.  a )  ->  (
 b  e.  N  /\  ( c  e.  b  ->  c  e.  a ) ) ) )
 
Theoremwopprc 27123 Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V )  <->  -. 
 1o  e.  { { { A } ,  (/) } ,  { { B } } } )
 
Theoremrpnnen3lem 27124* Lemma for rpnnen3 27125. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  (
 ( ( a  e. 
 RR  /\  b  e.  RR )  /\  a  < 
 b )  ->  { c  e.  QQ  |  c  < 
 a }  =/=  {
 c  e.  QQ  |  c  <  b } )
 
Theoremrpnnen3 27125 Dedekind cut injection of  RR into  ~P QQ. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  RR  ~<_  ~P QQ
 
18.17.39  More equivalents of the Axiom of Choice
 
Theoremaxac10 27126 Characterization of choice similar to dffin1-5 8014. (Contributed by Stefan O'Rear, 6-Jan-2015.)
 |-  (  ~~  " On )  =  _V
 
Theoremharinf 27127 The Hartogs number of an infinite set is at least  om. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  (
 ( S  e.  V  /\  -.  S  e.  Fin )  ->  om  C_  (har `  S ) )
 
Theoremwdom2d2 27128* Deduction for weak dominance by a cross product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  (
 ( ph  /\  x  e.  A )  ->  E. y  e.  B  E. z  e.  C  x  =  X )   =>    |-  ( ph  ->  A  ~<_*  ( B  X.  C ) )
 
Theoremttac 27129 Tarski's theorem about choice: infxpidm 8184 is equivalent to ax-ac 8085. (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 27130* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 6969, 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 27131* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 6969, 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 27132* Function value of the pw2f1o2 27131 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 27133* Membership in a mapped set under the pw2f1o2 27131 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 27134 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 27135 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 27136 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 27137 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 27138* 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 27139* 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 )
 
Theoreminisegn0 27140 Non-emptyness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)
 |-  ( A  e.  ran  F  <->  ( `' F " { A } )  =/= 
 (/) )
 
Theoremdnnumch1 27141* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 7657 (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 27142* 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 27143* 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 27144* 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 27145* 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 27146* Lemma for fnwe2 27150. 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 27147* Lemma for fnwe2 27150. 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 27148* Lemma for fnwe2 27150. 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 27149* Lemma for fnwe2 27150. 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 27150* 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 6231 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 27151* Lemma for dfac11 27160. 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 27152* Lemma for dfac11 27160. 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 27153* Lemma for dfac11 27160. 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 27154* Lemma for dfac11 27160. Limit case. Patch together well-orderings constructed so far using fnwe2 27150 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 27155* Lemma for dfac11 27160. 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 27156* Lemma for dfac11 27160. 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 27157* Lemma for dfac11 27160. 
( 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 ) )
 
Theoremsupeq123d 27158 Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
 |-  ( ph  ->  A  =  D )   &    |-  ( ph  ->  B  =  E )   &    |-  ( ph  ->  C  =  F )   =>    |-  ( ph  ->  sup ( A ,  B ,  C )  =  sup ( D ,  E ,  F ) )
 
Theoremaomclem8 27159* Lemma for dfac11 27160. 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 27160* The right hand side of this theorem (compare with ac4 8102), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 7306, 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 27161* 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 27162 Lemma for kelac2 27163 and dfac21 27164: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( S  e.  V  ->  (
 topGen `  { S ,  { ~P U. S } } )  e.  Comp )
 
Theoremkelac2 27163* 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 27164 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 ) )
 
18.17.40  Finitely generated left modules
 
Syntaxclfig 27165 Extend class notation with the class of finitely generated left modules.
 class LFinGen
 
Definitiondf-lfig 27166 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 27167* 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 27168* 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 27169* 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 27170 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 27171 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( M  e. LFinGen  ->  M  e.  LMod
 )
 
Theoremlsmfgcl 27172 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 )
 
18.17.41  Noetherian left modules I
 
Syntaxclnm 27173 Extend class notation with the class of Noetherian left modules.
 class LNoeM
 
Definitiondf-lnm 27174* 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 27175* 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 27176* 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 27177 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  ( M  e. LNoeM  ->  M  e.  LMod
 )
 
Theoremlnmlssfg 27178 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 27179 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 27180 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  ( M  e. LNoeM  ->  M  e. LFinGen )
 
Theoremkercvrlsm 27181 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 27182 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 27183 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 27184 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 27185 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 27186 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑚 
 S  ->  ( R  e. LNoeM  <->  S  e. LNoeM ) )
 
18.17.42  Addenda for structure powers
 
Theorempwssplit0 27187* Splitting for structure powers, part 0: restriction is a function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  U )   &    |-  Z  =  ( W  ^s  V )   &    |-  B  =  ( Base `  Y )   &    |-  C  =  ( Base `  Z )   &    |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( ( W  e.  T  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
 
Theorempwssplit1 27188* 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.)
 |-  Y  =  ( W  ^s  U )   &    |-  Z  =  ( W  ^s  V )   &    |-  B  =  ( Base `  Y )   &    |-  C  =  ( Base `  Z )   &    |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B -onto-> C )
 
Theorempwssplit2 27189* Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  U )   &    |-  Z  =  ( W  ^s  V )   &    |-  B  =  ( Base `  Y )   &    |-  C  =  ( Base `  Z )   &    |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
 
Theorempwssplit3 27190* Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  U )   &    |-  Z  =  ( W  ^s  V )   &    |-  B  =  ( Base `  Y )   &    |-  C  =  ( Base `  Z )   &    |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y LMHom  Z ) )
 
Theorempwssplit4 27191* 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 27192 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 27193 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  (/) )   =>    |-  ( W  e.  LMod 
 ->  Y  e. LNoeM )
 
Theorempwslnmlem1 27194* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  { i } )   =>    |-  ( W  e. LNoeM  ->  Y  e. LNoeM )
 
Theorempwslnmlem2 27195 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 27196 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 )
 
18.17.43  Direct sum of left modules
 
Syntaxcdsmm 27197 Class of module direct sum generator.
 class  (+)m
 
Definitiondf-dsmm 27198* 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  =  ( s  e.  _V ,  r  e.  _V  |->  ( ( s X_s r
 )s  { f  e.  X_ x  e.  dom  r ( Base `  ( r `  x ) )  |  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  e.  Fin
 } ) )
 
Theoremreldmdsmm 27199 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |-  Rel  dom  (+)m
 
Theoremdsmmval 27200* Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |-  B  =  { f  e.  ( Base `  ( S X_s R ) )  |  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin }   =>    |-  ( R  e.  V  ->  ( S  (+)m  R )  =  ( ( S
 X_s
 R )s  B ) )
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