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Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfreq12d 26301 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 26302 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 26303 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 26304* 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 26305* 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 26306 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 26307* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 7541 (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 26308* 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 26309* 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 26310* 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 26311* 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 26312* Lemma for fnwe2 26316. 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 26313* Lemma for fnwe2 26316. 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 26314* Lemma for fnwe2 26316. 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 26315* Lemma for fnwe2 26316. 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 26316* 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 6083 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 26317* Lemma for dfac11 26326. 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 26318* Lemma for dfac11 26326. 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 26319* Lemma for dfac11 26326. 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 26320* Lemma for dfac11 26326. Limit case. Patch together well-orderings constructed so far using fnwe2 26316 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 26321* Lemma for dfac11 26326. 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 26322* Lemma for dfac11 26326. 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 26323* Lemma for dfac11 26326. 
( 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 26324 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 26325* Lemma for dfac11 26326. 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 26326* The right hand side of this theorem (compare with ac4 7986), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 7190, 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 26327* 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 26328 Lemma for kelac2 26329 and dfac21 26330: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( S  e.  V  ->  (
 topGen `  { S ,  { ~P U. S } } )  e.  Comp )
 
Theoremkelac2 26329* 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 26330 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 ) )
 
16.15.40  Finitely generated left modules
 
Syntaxclfig 26331 Extend class notation with the class of finitely generated left modules.
 class LFinGen
 
Definitiondf-lfig 26332 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 26333* 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 26334* 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 26335* 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 26336 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 26337 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( M  e. LFinGen  ->  M  e.  LMod
 )
 
Theoremlsmfgcl 26338 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 )
 
16.15.41  Noetherian left modules I
 
Syntaxclnm 26339 Extend class notation with the class of Noetherian left modules.
 class LNoeM
 
Definitiondf-lnm 26340* 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 26341* 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 26342* 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 26343 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  ( M  e. LNoeM  ->  M  e.  LMod
 )
 
Theoremlnmlssfg 26344 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 26345 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 26346 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  ( M  e. LNoeM  ->  M  e. LFinGen )
 
Theoremkercvrlsm 26347 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 26348 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 26349 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 26350 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 26351 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 26352 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑚 
 S  ->  ( R  e. LNoeM  <->  S  e. LNoeM ) )
 
16.15.42  Addenda for structure powers
 
Theorempwssplit0 26353* 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 26354* 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 26355* 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 26356* 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 26357* 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 26358 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 26359 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  (/) )   =>    |-  ( W  e.  LMod 
 ->  Y  e. LNoeM )
 
Theorempwslnmlem1 26360* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  { i } )   =>    |-  ( W  e. LNoeM  ->  Y  e. LNoeM )
 
Theorempwslnmlem2 26361 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 26362 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 )
 
16.15.43  Direct sum of left modules
 
Syntaxcdsmm 26363 Class of module direct sum generator.
 class  (+)m
 
Definitiondf-dsmm 26364* 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 26365 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |-  Rel  dom  (+)m
 
Theoremdsmmval 26366* 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 ) )
 
Theoremdsmmbase 26367* Base set 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  ->  B  =  ( Base `  ( S  (+)m  R ) ) )
 
Theoremdsmmval2 26368 Self-referential definition of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  B  =  ( Base `  ( S  (+)m  R ) )   =>    |-  ( S  (+)m  R )  =  ( ( S X_s R )s  B )
 
Theoremdsmmbas2 26369* Base set of the direct sum module using the fndmin 5484 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  P  =  ( S X_s R )   &    |-  B  =  {
 f  e.  ( Base `  P )  |  dom  (  f  \  ( 0g 
 o.  R ) )  e.  Fin }   =>    |-  ( ( R  Fn  I  /\  I  e.  V )  ->  B  =  ( Base `  ( S  (+)m  R ) ) )
 
Theoremdsmmfi 26370 For finite products, the direct sum is just the module product. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  (
 ( R  Fn  I  /\  I  e.  Fin )  ->  ( S  (+)m  R )  =  ( S X_s R ) )
 
Theoremdsmmelbas 26371* Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  P  =  ( S X_s R )   &    |-  C  =  ( S  (+)m  R )   &    |-  B  =  (
 Base `  P )   &    |-  H  =  ( Base `  C )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  Fn  I )   =>    |-  ( ph  ->  ( X  e.  H  <->  ( X  e.  B  /\  { a  e.  I  |  ( X `
  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
 ) )
 
Theoremdsmm0cl 26372 The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  P  =  ( S X_s R )   &    |-  H  =  (
 Base `  ( S  (+)m  R ) )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Mnd )   &    |-  .0.  =  ( 0g `  P )   =>    |-  ( ph  ->  .0.  e.  H )
 
Theoremdsmmacl 26373 The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  P  =  ( S X_s R )   &    |-  H  =  (
 Base `  ( S  (+)m  R ) )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Mnd )   &    |-  ( ph  ->  J  e.  H )   &    |-  ( ph  ->  K  e.  H )   &    |- 
 .+  =  ( +g  `  P )   =>    |-  ( ph  ->  ( J  .+  K )  e.  H )
 
Theoremprdsinvgd2 26374 Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  Y  =  ( S X_s R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Grp )   &    |-  B  =  (
 Base `  Y )   &    |-  N  =  ( inv g `  Y )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  J  e.  I
 )   =>    |-  ( ph  ->  (
 ( N `  X ) `  J )  =  ( ( inv g `  ( R `  J ) ) `  ( X `  J ) ) )
 
Theoremdsmmsubg 26375 The finite hull of a product of groups is additionally closed under negation and thus is a subgroup of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  P  =  ( S X_s R )   &    |-  H  =  (
 Base `  ( S  (+)m  R ) )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Grp )   =>    |-  ( ph  ->  H  e.  (SubGrp `  P )
 )
 
Theoremdsmmlss 26376* The finite hull of a product of modules is additionally closed under scalar multiplication and thus is a linear subspace of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  R : I --> LMod )   &    |-  (
 ( ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x ) )  =  S )   &    |-  P  =  ( S
 X_s
 R )   &    |-  U  =  (
 LSubSp `  P )   &    |-  H  =  ( Base `  ( S  (+)m  R ) )   =>    |-  ( ph  ->  H  e.  U )
 
Theoremdsmmlmod 26377* The direct sum of a family of modules is a module. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  R : I --> LMod )   &    |-  (
 ( ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x ) )  =  S )   &    |-  C  =  ( S 
 (+)m  R )   =>    |-  ( ph  ->  C  e.  LMod )
 
16.15.44  Free modules
 
Syntaxcfrlm 26378 Class of free module generator.
 class freeLMod
 
Syntaxcuvc 26379 Class of basic unit vectors for an explicit free module.
 class unitVec
 
Definitiondf-frlm 26380* The  i-dimensional free module over a ring  r is the product of  i-many copies of the ring with componentwise addition and multiplication. If  i is infinite, the allowed vectors are restricted to those with finitely many nonzero coordinates; this ensures that the resulting module is actually spanned by its unit vectors. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- freeLMod  =  ( r  e.  _V ,  i  e.  _V  |->  ( r 
 (+)m  ( i  X.  {
 (ringLMod `  r ) }
 ) ) )
 
Definitiondf-uvc 26381*  ( ( R unitVec  I ) `  i
) is the unit vector in 
( R freeLMod  I ) along the  i axis. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- unitVec  =  ( r  e.  _V ,  i  e.  _V  |->  ( j  e.  i  |->  ( k  e.  i  |->  if (
 k  =  j ,  ( 1r `  r
 ) ,  ( 0g
 `  r ) ) ) ) )
 
Theoremfrlmval 26382 Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   =>    |-  (
 ( R  e.  V  /\  I  e.  W )  ->  F  =  ( R  (+)m  ( I  X.  {
 (ringLMod `  R ) }
 ) ) )
 
Theoremfrlmlmod 26383 The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  W ) 
 ->  F  e.  LMod )
 
Theoremfrlmpws 26384 The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  F )   =>    |-  (
 ( R  e.  V  /\  I  e.  W )  ->  F  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
 
Theoremfrlmlss 26385 The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  F )   &    |-  U  =  ( LSubSp `  ( (ringLMod `  R )  ^s  I )
 )   =>    |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  B  e.  U )
 
Theoremfrlmpwsfi 26386 The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   =>    |-  (
 ( R  e.  V  /\  I  e.  Fin )  ->  F  =  ( (ringLMod `  R )  ^s  I ) )
 
Theoremfrlmsca 26387 The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   =>    |-  (
 ( R  e.  V  /\  I  e.  W )  ->  R  =  (Scalar `  F ) )
 
Theoremfrlm0 26388 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 26385). (Contributed by Stefan O'Rear, 4-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( I  X.  {  .0.  } )  =  ( 0g `  F ) )
 
Theoremfrlmbas 26389* Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  N  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  { k  e.  ( N  ^m  I
 )  |  ( `' k " ( _V  \  {  .0.  } )
 )  e.  Fin }   =>    |-  (
 ( R  e.  V  /\  I  e.  W )  ->  B  =  (
 Base `  F ) )
 
Theoremfrlmelbas 26390 Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  N  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  F )   =>    |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( X  e.  B 
 <->  ( X  e.  ( N  ^m  I )  /\  ( `' X " ( _V  \  {  .0.  } )
 )  e.  Fin )
 ) )
 
Theoremfrlmrcl 26391 If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  F )   =>    |-  ( X  e.  B  ->  R  e.  _V )
 
Theoremfrlmbassup 26392 Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  F )   =>    |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( `' X " ( _V  \  {  .0.  } )
 )  e.  Fin )
 
Theoremfrlmbasmap 26393 Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  N  =  ( Base `  R )   &    |-  B  =  ( Base `  F )   =>    |-  (
 ( I  e.  W  /\  X  e.  B ) 
 ->  X  e.  ( N 
 ^m  I ) )
 
Theoremfrlmbasf 26394 Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  N  =  ( Base `  R )   &    |-  B  =  ( Base `  F )   =>    |-  (
 ( I  e.  W  /\  X  e.  B ) 
 ->  X : I --> N )
 
Theoremfrlmplusgval 26395 Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  Y )   =>    |-  ( ph  ->  ( F  .+b  G )  =  ( F  o F  .+  G ) )
 
Theoremfrlmvscafval 26396 Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   &    |-  K  =  ( Base `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  B )   &    |-  .xb  =  ( .s `  Y )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ph  ->  ( A  .xb  X )  =  ( ( I  X.  { A } )  o F  .x.  X )
 )
 
Theoremfrlmvscaval 26397 Scalar multiplication in a free module at a coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   &    |-  K  =  ( Base `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  J  e.  I
 )   &    |-  .xb  =  ( .s `  Y )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ph  ->  ( ( A  .xb  X ) `
  J )  =  ( A  .x.  ( X `  J ) ) )
 
Theoremfrlmgsum 26398* Finite commutative sums in a free module are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 5-Jul-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  J  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ( ph  /\  y  e.  J )  ->  ( x  e.  I  |->  U )  e.  B )   &    |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) "
 ( _V  \  {  .0.  } ) )  e. 
 Fin )   =>    |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
 
Theoremuvcfval 26399* Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  W )  ->  U  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) )
 
Theoremuvcval 26400* Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I ) 
 ->  ( U `  J )  =  ( k  e.  I  |->  if (
 k  =  J ,  .1.  ,  .0.  ) ) )
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