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Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1ocpbl 13301 An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  F : V -1-1-onto-> X )   =>    |-  ( ( ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V ) )  ->  ( (
 ( F `  A )  =  ( F `  C )  /\  ( F `  B )  =  ( F `  D ) )  ->  ( F `
  ( A  .+  B ) )  =  ( F `  ( C  .+  D ) ) ) )
 
Theoremf1ovscpbl 13302 An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  ( ph  ->  F : V -1-1-onto-> X )   =>    |-  ( ( ph  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( F `  B )  =  ( F `  C )  ->  ( F `  ( A  .+  B ) )  =  ( F `
  ( A  .+  C ) ) ) )
 
Theoremf1olecpbl 13303 An injection is compatible with any relations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  F : V -1-1-onto-> X )   =>    |-  ( ( ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V ) )  ->  ( (
 ( F `  A )  =  ( F `  C )  /\  ( F `  B )  =  ( F `  D ) )  ->  ( A N B  <->  C N D ) ) )
 
Theoremimasaddfnlem 13304* The image structure operation is a function if the original operation is compatible with the function. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  .xb 
 =  U_ p  e.  V  U_ q  e.  V  { <.
 <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p  .x.  q ) ) >. } )   =>    |-  ( ph  ->  .xb  Fn  ( B  X.  B ) )
 
Theoremimasaddvallem 13305* The operation of an image structure is defined to distribute over the mapping function. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  .xb 
 =  U_ p  e.  V  U_ q  e.  V  { <.
 <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p  .x.  q ) ) >. } )   =>    |-  ( ( ph  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( F `  X ) 
 .xb  ( F `  Y ) )  =  ( F `  ( X  .x.  Y ) ) )
 
Theoremimasaddflem 13306* The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  .xb 
 =  U_ p  e.  V  U_ q  e.  V  { <.
 <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p  .x.  q ) ) >. } )   &    |-  ( ( ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( p 
 .x.  q )  e.  V )   =>    |-  ( ph  ->  .xb  : ( B  X.  B ) --> B )
 
Theoremimasaddfn 13307* The image structure's group operation is a function. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
 |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  U  =  ( F  "s  R ) )   &    |-  ( ph  ->  V  =  ( Base `  R ) )   &    |-  ( ph  ->  R  e.  Z )   &    |-  .x.  =  ( +g  `  R )   &    |-  .xb  =  ( +g  `  U )   =>    |-  ( ph  ->  .xb  Fn  ( B  X.  B ) )
 
Theoremimasaddval 13308* The value of an image structure's group operation. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  U  =  ( F  "s  R ) )   &    |-  ( ph  ->  V  =  ( Base `  R ) )   &    |-  ( ph  ->  R  e.  Z )   &    |-  .x.  =  ( +g  `  R )   &    |-  .xb  =  ( +g  `  U )   =>    |-  (
 ( ph  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( F `
  X )  .xb  ( F `  Y ) )  =  ( F `
  ( X  .x.  Y ) ) )
 
Theoremimasaddf 13309* The image structure's group operation is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  U  =  ( F  "s  R ) )   &    |-  ( ph  ->  V  =  ( Base `  R ) )   &    |-  ( ph  ->  R  e.  Z )   &    |-  .x.  =  ( +g  `  R )   &    |-  .xb  =  ( +g  `  U )   &    |-  (
 ( ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( p  .x.  q )  e.  V )   =>    |-  ( ph  ->  .xb 
 : ( B  X.  B ) --> B )
 
Theoremimasmulfn 13310* The image structure's ring multiplication is a function. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  U  =  ( F  "s  R ) )   &    |-  ( ph  ->  V  =  ( Base `  R ) )   &    |-  ( ph  ->  R  e.  Z )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  U )   =>    |-  ( ph  ->  .xb  Fn  ( B  X.  B ) )
 
Theoremimasmulval 13311* The value of an image structure's ring multiplication. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  U  =  ( F  "s  R ) )   &    |-  ( ph  ->  V  =  ( Base `  R ) )   &    |-  ( ph  ->  R  e.  Z )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  U )   =>    |-  ( ( ph  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( F `  X ) 
 .xb  ( F `  Y ) )  =  ( F `  ( X  .x.  Y ) ) )
 
Theoremimasmulf 13312* The image structure's ring multiplication is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  U  =  ( F  "s  R ) )   &    |-  ( ph  ->  V  =  ( Base `  R ) )   &    |-  ( ph  ->  R  e.  Z )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  U )   &    |-  ( ( ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( p 
 .x.  q )  e.  V )   =>    |-  ( ph  ->  .xb  : ( B  X.  B ) --> B )
 
Theoremimasvscafn 13313* The image structure's scalar multiplication is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  G  =  (Scalar `  R )   &    |-  K  =  ( Base `  G )   &    |-  .x.  =  ( .s `  R )   &    |-  .xb 
 =  ( .s `  U )   &    |-  ( ( ph  /\  ( p  e.  K  /\  a  e.  V  /\  q  e.  V ) )  ->  ( ( F `  a )  =  ( F `  q )  ->  ( F `
  ( p  .x.  a ) )  =  ( F `  ( p  .x.  q ) ) ) )   =>    |-  ( ph  ->  .xb  Fn  ( K  X.  B ) )
 
Theoremimasvscaval 13314* The value of an image structure's scalar multiplication. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  G  =  (Scalar `  R )   &    |-  K  =  ( Base `  G )   &    |-  .x.  =  ( .s `  R )   &    |-  .xb 
 =  ( .s `  U )   &    |-  ( ( ph  /\  ( p  e.  K  /\  a  e.  V  /\  q  e.  V ) )  ->  ( ( F `  a )  =  ( F `  q )  ->  ( F `
  ( p  .x.  a ) )  =  ( F `  ( p  .x.  q ) ) ) )   =>    |-  ( ( ph  /\  X  e.  K  /\  Y  e.  V )  ->  ( X 
 .xb  ( F `  Y ) )  =  ( F `  ( X  .x.  Y ) ) )
 
Theoremimasvscaf 13315* The image structure's scalar multiplication is closed in the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  G  =  (Scalar `  R )   &    |-  K  =  ( Base `  G )   &    |-  .x.  =  ( .s `  R )   &    |-  .xb 
 =  ( .s `  U )   &    |-  ( ( ph  /\  ( p  e.  K  /\  a  e.  V  /\  q  e.  V ) )  ->  ( ( F `  a )  =  ( F `  q )  ->  ( F `
  ( p  .x.  a ) )  =  ( F `  ( p  .x.  q ) ) ) )   &    |-  ( ( ph  /\  ( p  e.  K  /\  q  e.  V ) )  ->  ( p 
 .x.  q )  e.  V )   =>    |-  ( ph  ->  .xb  : ( K  X.  B ) --> B )
 
Theoremimasless 13316 The order relation defined on an image set is a subset of the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  .<_  =  ( le `  U )   =>    |-  ( ph  ->  .<_  C_  ( B  X.  B ) )
 
Theoremimasleval 13317* The value of the image structure's ordering when the order is compatible with the mapping function. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  .<_  =  ( le `  U )   &    |-  N  =  ( le `  R )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( c  e.  V  /\  d  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  c
 )  /\  ( F `  b )  =  ( F `  d ) )  ->  ( a N b  <->  c N d ) ) )   =>    |-  ( ( ph  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( F `  X )  .<_  ( F `
  Y )  <->  X N Y ) )
 
Theoremdivsval 13318* Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  F  =  ( x  e.  V  |->  [ x ]  .~  )   &    |-  ( ph  ->  .~  e.  W )   &    |-  ( ph  ->  R  e.  Z )   =>    |-  ( ph  ->  U  =  ( F  "s  R )
 )
 
Theoremdivslem 13319* The function in divsval 13318 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  F  =  ( x  e.  V  |->  [ x ]  .~  )   &    |-  ( ph  ->  .~  e.  W )   &    |-  ( ph  ->  R  e.  Z )   =>    |-  ( ph  ->  F : V -onto-> ( V /.  .~  ) )
 
Theoremdivsin 13320 Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .~  e.  W )   &    |-  ( ph  ->  R  e.  Z )   &    |-  ( ph  ->  (  .~  " V )  C_  V )   =>    |-  ( ph  ->  U  =  ( R  /.s  (  .~  i^i  ( V  X.  V ) ) ) )
 
Theoremdivsbas 13321 Base set of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .~  e.  W )   &    |-  ( ph  ->  R  e.  Z )   =>    |-  ( ph  ->  ( V /.  .~  )  =  ( Base `  U )
 )
 
Theoremdivssca 13322 The scalar field of a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .~  e.  W )   &    |-  ( ph  ->  R  e.  Z )   &    |-  K  =  (Scalar `  R )   =>    |-  ( ph  ->  K  =  (Scalar `  U ) )
 
Theoremdivsfval 13323* Value of the function in divsval 13318. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  V  e.  _V )   &    |-  F  =  ( x  e.  V  |->  [ x ]  .~  )   =>    |-  ( ph  ->  ( F `  A )  =  [ A ]  .~  )
 
Theoremercpbllem 13324* Lemma for ercpbl 13325. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  V  e.  _V )   &    |-  F  =  ( x  e.  V  |->  [ x ]  .~  )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  (
 ( F `  A )  =  ( F `  B )  <->  A  .~  B ) )
 
Theoremercpbl 13325* Translate the function compatiblity relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  V  e.  _V )   &    |-  F  =  ( x  e.  V  |->  [ x ]  .~  )   &    |-  (
 ( ph  /\  ( a  e.  V  /\  b  e.  V ) )  ->  ( a  .+  b )  e.  V )   &    |-  ( ph  ->  ( ( A 
 .~  C  /\  B  .~  D )  ->  ( A  .+  B )  .~  ( C  .+  D ) ) )   =>    |-  ( ( ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V ) )  ->  ( (
 ( F `  A )  =  ( F `  C )  /\  ( F `  B )  =  ( F `  D ) )  ->  ( F `
  ( A  .+  B ) )  =  ( F `  ( C  .+  D ) ) ) )
 
Theoremerlecpbl 13326* Translate the relation compatiblity relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  V  e.  _V )   &    |-  F  =  ( x  e.  V  |->  [ x ]  .~  )   &    |-  ( ph  ->  ( ( A 
 .~  C  /\  B  .~  D )  ->  ( A N B  <->  C N D ) ) )   =>    |-  ( ( ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V ) )  ->  ( (
 ( F `  A )  =  ( F `  C )  /\  ( F `  B )  =  ( F `  D ) )  ->  ( A N B  <->  C N D ) ) )
 
Theoremdivsaddvallem 13327* Value of an operation defined on a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  R  e.  Z )   &    |-  ( ph  ->  ( ( a 
 .~  p  /\  b  .~  q )  ->  (
 a  .x.  b )  .~  ( p  .x.  q
 ) ) )   &    |-  (
 ( ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( p  .x.  q )  e.  V )   &    |-  F  =  ( x  e.  V  |->  [ x ]  .~  )   &    |-  ( ph  ->  .xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `
  p ) ,  ( F `  q
 ) >. ,  ( F `
  ( p  .x.  q ) ) >. } )   =>    |-  ( ( ph  /\  X  e.  V  /\  Y  e.  V )  ->  ( [ X ]  .~  .xb  [ Y ]  .~  )  =  [
 ( X  .x.  Y ) ]  .~  )
 
Theoremdivsaddflem 13328* The operation of a quotient structure is a function. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  R  e.  Z )   &    |-  ( ph  ->  ( ( a 
 .~  p  /\  b  .~  q )  ->  (
 a  .x.  b )  .~  ( p  .x.  q
 ) ) )   &    |-  (
 ( ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( p  .x.  q )  e.  V )   &    |-  F  =  ( x  e.  V  |->  [ x ]  .~  )   &    |-  ( ph  ->  .xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `
  p ) ,  ( F `  q
 ) >. ,  ( F `
  ( p  .x.  q ) ) >. } )   =>    |-  ( ph  ->  .xb  : ( ( V /.  .~  )  X.  ( V /.  .~  ) ) --> ( V
 /.  .~  ) )
 
Theoremdivsaddval 13329* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  R  e.  Z )   &    |-  ( ph  ->  ( ( a 
 .~  p  /\  b  .~  q )  ->  (
 a  .x.  b )  .~  ( p  .x.  q
 ) ) )   &    |-  (
 ( ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( p  .x.  q )  e.  V )   &    |-  .x.  =  ( +g  `  R )   &    |-  .xb  =  ( +g  `  U )   =>    |-  (
 ( ph  /\  X  e.  V  /\  Y  e.  V )  ->  ( [ X ]  .~  .xb  [ Y ]  .~  )  =  [
 ( X  .x.  Y ) ]  .~  )
 
Theoremdivsaddf 13330* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  R  e.  Z )   &    |-  ( ph  ->  ( ( a 
 .~  p  /\  b  .~  q )  ->  (
 a  .x.  b )  .~  ( p  .x.  q
 ) ) )   &    |-  (
 ( ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( p  .x.  q )  e.  V )   &    |-  .x.  =  ( +g  `  R )   &    |-  .xb  =  ( +g  `  U )   =>    |-  ( ph  ->  .xb  : ( ( V /.  .~  )  X.  ( V /.  .~  ) ) --> ( V
 /.  .~  ) )
 
Theoremdivsmulval 13331* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  R  e.  Z )   &    |-  ( ph  ->  ( ( a 
 .~  p  /\  b  .~  q )  ->  (
 a  .x.  b )  .~  ( p  .x.  q
 ) ) )   &    |-  (
 ( ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( p  .x.  q )  e.  V )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  U )   =>    |-  ( ( ph  /\  X  e.  V  /\  Y  e.  V )  ->  ( [ X ]  .~  .xb  [ Y ]  .~  )  =  [
 ( X  .x.  Y ) ]  .~  )
 
Theoremdivsmulf 13332* The base set of an image structure. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  R  e.  Z )   &    |-  ( ph  ->  ( ( a 
 .~  p  /\  b  .~  q )  ->  (
 a  .x.  b )  .~  ( p  .x.  q
 ) ) )   &    |-  (
 ( ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( p  .x.  q )  e.  V )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  U )   =>    |-  ( ph  ->  .xb  : ( ( V /.  .~  )  X.  ( V /.  .~  ) ) --> ( V
 /.  .~  ) )
 
Theoremxpsc 13333 A short expression for the pair function mapping  0 to  A and  1 to  B. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  `' ( { A }  +c  { B } )  =  ( ( { (/) }  X.  { A } )  u.  ( { 1o }  X.  { B } )
 )
 
Theoremxpscg 13334 A short expression for the pair function mapping  0 to  A and  1 to  B. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  `' ( { A }  +c  { B } )  =  { <.
 (/) ,  A >. , 
 <. 1o ,  B >. } )
 
Theoremxpscfn 13335 The pair function is a function on 
2o  =  { (/) ,  1o }. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  `' ( { A }  +c  { B } )  Fn  2o )
 
Theoremxpsc0 13336 The pair function maps  0 to  A. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( A  e.  V  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
 
Theoremxpsc1 13337 The pair function maps  1 to  B. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B } ) `  1o )  =  B )
 
Theoremxpscfv 13338 The value of the pair function at an element of  2o. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  2o )  ->  ( `' ( { A }  +c  { B } ) `  C )  =  if ( C  =  (/) ,  A ,  B ) )
 
Theoremxpsfrnel 13339* Elementhood in the target space of the function  F appearing in xpsval 13348. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( G  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <->  ( G  Fn  2o  /\  ( G `  (/) )  e.  A  /\  ( G `  1o )  e.  B ) )
 
Theoremxpsfeq 13340 A function on  2o is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( G  Fn  2o  ->  `' ( { ( G `
  (/) ) }  +c  { ( G `  1o ) } )  =  G )
 
Theoremxpsfrnel2 13341* Elementhood in the target space of the function  F appearing in xpsval 13348. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  ( `' ( { X }  +c  { Y } )  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <->  ( X  e.  A  /\  Y  e.  B ) )
 
Theoremxpscf 13342 Equivalent condition for the pair function to be a proper function on  A. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( `' ( { X }  +c  { Y } ) : 2o --> A 
 <->  ( X  e.  A  /\  Y  e.  A ) )
 
Theoremxpsfval 13343* The value of the function appearing in xpsval 13348. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } )
 )   =>    |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( X F Y )  =  `' ( { X }  +c  { Y } ) )
 
Theoremxpsff1o 13344* The function appearing in xpsval 13348 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair  2o  =  { (/)
,  1o }. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } )
 )   =>    |-  F : ( A  X.  B ) -1-1-onto-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
 
Theoremxpsfrn 13345* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } )
 )   =>    |- 
 ran  F  =  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
 
Theoremxpsfrn2 13346* A short expression for the indexed cartesian product on two indexes. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } )
 )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  F  =  X_ k  e.  2o  ( `' ( { A }  +c  { B } ) `  k ) )
 
Theoremxpsff1o2 13347* The function appearing in xpsval 13348 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair  2o  =  { (/)
,  1o }. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } )
 )   =>    |-  F : ( A  X.  B ) -1-1-onto-> ran  F
 
Theoremxpsval 13348* Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
 y } ) )   &    |-  G  =  (Scalar `  R )   &    |-  U  =  ( G
 X_s `' ( { R }  +c  { S } )
 )   =>    |-  ( ph  ->  T  =  ( `' F  "s  U ) )
 
Theoremxpslem 13349* The indexed structure product that appears in xpsval 13348 has the same base as the target of the function  F. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  {
 y } ) )   &    |-  G  =  (Scalar `  R )   &    |-  U  =  ( G
 X_s `' ( { R }  +c  { S } )
 )   =>    |-  ( ph  ->  ran  F  =  ( Base `  U )
 )
 
Theoremxpsbas 13350 The base set of the binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   =>    |-  ( ph  ->  ( X  X.  Y )  =  ( Base `  T )
 )
 
Theoremxpsaddlem 13351* Lemma for xpsadd 13352 and xpsmul 13353. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   &    |-  ( ph  ->  ( A  .x.  C )  e.  X )   &    |-  ( ph  ->  ( B  .X. 
 D )  e.  Y )   &    |- 
 .x.  =  ( E `  R )   &    |-  .X.  =  ( E `  S )   &    |-  .xb  =  ( E `  T )   &    |-  F  =  ( x  e.  X ,  y  e.  Y  |->  `' ( { x }  +c  { y } )
 )   &    |-  U  =  ( (Scalar `  R ) X_s `' ( { R }  +c  { S } )
 )   &    |-  ( ( ph  /\  `' ( { A }  +c  { B } )  e. 
 ran  F  /\  `' ( { C }  +c  { D } )  e.  ran  F )  ->  ( ( `' F `  `' ( { A }  +c  { B } ) )  .xb  ( `' F `  `' ( { C }  +c  { D } ) ) )  =  ( `' F `  ( `' ( { A }  +c  { B } ) ( E `
  U ) `' ( { C }  +c  { D } )
 ) ) )   &    |-  (
 ( `' ( { R }  +c  { S } )  Fn  2o  /\  `' ( { A }  +c  { B } )  e.  ( Base `  U )  /\  `' ( { C }  +c  { D } )  e.  ( Base `  U )
 )  ->  ( `' ( { A }  +c  { B } ) ( E `  U ) `' ( { C }  +c  { D } )
 )  =  ( k  e.  2o  |->  ( ( `' ( { A }  +c  { B } ) `  k ) ( E `
  ( `' ( { R }  +c  { S } ) `  k
 ) ) ( `' ( { C }  +c  { D } ) `  k ) ) ) )   =>    |-  ( ph  ->  ( <. A ,  B >.  .xb  <. C ,  D >. )  =  <. ( A  .x.  C ) ,  ( B 
 .X.  D ) >. )
 
Theoremxpsadd 13352 Value of the addition operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   &    |-  ( ph  ->  ( A  .x.  C )  e.  X )   &    |-  ( ph  ->  ( B  .X. 
 D )  e.  Y )   &    |- 
 .x.  =  ( +g  `  R )   &    |-  .X.  =  ( +g  `  S )   &    |-  .xb  =  ( +g  `  T )   =>    |-  ( ph  ->  ( <. A ,  B >.  .xb  <. C ,  D >. )  =  <. ( A 
 .x.  C ) ,  ( B  .X.  D ) >. )
 
Theoremxpsmul 13353 Value of the multiplication operation in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   &    |-  ( ph  ->  ( A  .x.  C )  e.  X )   &    |-  ( ph  ->  ( B  .X. 
 D )  e.  Y )   &    |- 
 .x.  =  ( .r `  R )   &    |-  .X.  =  ( .r `  S )   &    |-  .xb  =  ( .r `  T )   =>    |-  ( ph  ->  ( <. A ,  B >.  .xb  <. C ,  D >. )  =  <. ( A  .x.  C ) ,  ( B  .X.  D ) >. )
 
Theoremxpssca 13354 Value of the scalar field of a binary structure product. For concreteness we choose the scalar field to match the left argument, but in most cases where this slot is meaningful both factors will have the same scalar field, so that it doesn't matter which factor is chosen. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  G  =  (Scalar `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   =>    |-  ( ph  ->  G  =  (Scalar `  T )
 )
 
Theoremxpsvsca 13355 Value of the scalar multiplication function in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  G  =  (Scalar `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  X  =  ( Base `  R )   &    |-  Y  =  (
 Base `  S )   &    |-  K  =  ( Base `  G )   &    |-  .x.  =  ( .s `  R )   &    |-  .X. 
 =  ( .s `  S )   &    |-  .xb  =  ( .s `  T )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  Y )   &    |-  ( ph  ->  ( A  .x.  B )  e.  X )   &    |-  ( ph  ->  ( A  .X. 
 C )  e.  Y )   =>    |-  ( ph  ->  ( A  .xb  <. B ,  C >. )  =  <. ( A 
 .x.  B ) ,  ( A  .X.  C ) >. )
 
Theoremxpsless 13356 Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  .<_  =  ( le `  T )   =>    |-  ( ph  ->  .<_  C_  (
 ( X  X.  Y )  X.  ( X  X.  Y ) ) )
 
Theoremxpsle 13357 Value of the ordering in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  T  =  ( R  X.s  S )   &    |-  X  =  (
 Base `  R )   &    |-  Y  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  S  e.  W )   &    |-  .<_  =  ( le `  T )   &    |-  M  =  ( le `  R )   &    |-  N  =  ( le `  S )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   =>    |-  ( ph  ->  (
 <. A ,  B >.  .<_  <. C ,  D >.  <->  ( A M C  /\  B N D ) ) )
 
7.2  Moore spaces
 
Syntaxcmre 13358 The class of Moore systems.
 class Moore
 
Syntaxcmrc 13359 The class function generating Moore closures.
 class mrCls
 
Syntaxcacs 13360 The class of algebraic closure (Moore) systems.
 class ACS
 
Definitiondf-mre 13361* Define a Moore collection, which is a family of subsets of a base set which preserve arbitrary intersection. Elements of a Moore collection are termed closed; Moore collections generalize the notion of closedness from topologies (cldmre 16647) and vector spaces (lssmre 15558) to the most general setting in which such concepts make sense. Definition of Moore collection of sets in [Schechter] p. 78.

See ismre 13364, mresspw 13366, mre1cl 13368 and mreintcl 13369 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13374); as such the disjoint union of all Moore collections is sometimes considered as  U. ran Moore, justified by mreunirn 13375. (Contributed by Stefan O'Rear, 30-Jan-2015.)

 |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }
 )
 
Definitiondf-mrc 13362* Define the Moore closure of a generating set, which is the smallest closed set containing all generating elements. Definition of Moore closure in [Schechter] p. 79. This generalizes topological closure (mrccls 16648) and linear span (mrclsp 15581). (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- mrCls  =  ( c  e.  U. ran Moore 
 |->  ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) )
 
Definitiondf-acs 13363* An important subclass of Moore systems are those which can be interpreted as closure under some collection of operators of finite arity (the collection itself is not required to be finite). These are termed algebraic closure systems; similar to definition (A) of an algebraic closure system in [Schechter] p. 84, but to avoid the complexity of an arbitrary mixed collection of functions of various arities (especially if the axiom of infinity omex 7228 is to be avoided), we consider a single function defined on finite sets instead. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |- ACS 
 =  ( x  e. 
 _V  |->  { c  e.  (Moore `  x )  |  E. f ( f : ~P x --> ~P x  /\  A. s  e.  ~P  x ( s  e.  c  <->  U. ( f "
 ( ~P s  i^i 
 Fin ) )  C_  s ) ) }
 )
 
Theoremismre 13364* Property of being a Moore collection on some base set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e. 
 ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
 
Theoremfnmre 13365 The Moore collection generator is a well-behaved function. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |- Moore  Fn  _V
 
Theoremmresspw 13366 A Moore collection is a subset of the power of the base set; each closed subset of the system is actually a subset of the base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  ->  C  C_ 
 ~P X )
 
Theoremmress 13367 A Moore-closed subset is a subset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  S  e.  C )  ->  S  C_  X )
 
Theoremmre1cl 13368 In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  ->  X  e.  C )
 
Theoremmreintcl 13369 A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C  /\  S  =/= 
 (/) )  ->  |^| S  e.  C )
 
Theoremmreiincl 13370* A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  I  =/=  (/)  /\  A. y  e.  I  S  e.  C )  ->  |^|_ y  e.  I  S  e.  C )
 
Theoremmrerintcl 13371 The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  S  C_  C )  ->  ( X  i^i  |^| S )  e.  C )
 
Theoremmreriincl 13372* The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  A. y  e.  I  S  e.  C )  ->  ( X  i^i  |^|_ y  e.  I  S )  e.  C )
 
Theoremmreincl 13373 Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C  /\  B  e.  C )  ->  ( A  i^i  B )  e.  C )
 
Theoremmreuni 13374 Since the entire base set of a Moore collection is the greatest element of it, the base set can be recovered from a Moore collection by set union. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  ->  U. C  =  X )
 
Theoremmreunirn 13375 Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( C  e.  U. ran Moore  <->  C  e.  (Moore `  U. C ) )
 
Theoremismred 13376* Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( ph  ->  C  C_ 
 ~P X )   &    |-  ( ph  ->  X  e.  C )   &    |-  ( ( ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C )   =>    |-  ( ph  ->  C  e.  (Moore `  X )
 )
 
Theoremismred2 13377* Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( ph  ->  C  C_ 
 ~P X )   &    |-  (
 ( ph  /\  s  C_  C )  ->  ( X  i^i  |^| s )  e.  C )   =>    |-  ( ph  ->  C  e.  (Moore `  X )
 )
 
Theoremmremre 13378 The Moore collections of subsets of a space, viewed as a kind of subset of the power set, form a Moore collection in their own right on the power set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( X  e.  V  ->  (Moore `  X )  e.  (Moore `  ~P X ) )
 
Theoremsubmre 13379 The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  ( ( C  e.  (Moore `  X )  /\  A  e.  C )  ->  ( C  i^i  ~P A )  e.  (Moore `  A ) )
 
7.2.1  Moore closures
 
Theoremmrcflem 13380* The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  ( C  e.  (Moore `  X )  ->  ( x  e.  ~P X  |->  |^|
 { s  e.  C  |  x  C_  s }
 ) : ~P X --> C )
 
Theoremfnmrc 13381 Moore-closure is a well behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- mrCls  Fn  U. ran Moore
 
Theoremmrcfval 13382* Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
 
Theoremmrcf 13383 The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  F : ~P X --> C )
 
Theoremmrcval 13384* Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  =  |^| { s  e.  C  |  U  C_  s } )
 
Theoremmrccl 13385 The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  U )  e.  C )
 
Theoremmrcsncl 13386 The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  e.  X )  ->  ( F `  { U } )  e.  C )
 
Theoremmrcid 13387 The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  e.  C )  ->  ( F `  U )  =  U )
 
Theoremmrcssv 13388 The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  ( F `  U )  C_  X )
 
Theoremmrcidb 13389 A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  ( U  e.  C  <->  ( F `  U )  =  U ) )
 
Theoremmrcss 13390 Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  C_  X )  ->  ( F `  U )  C_  ( F `  V ) )
 
Theoremmrcssid 13391 The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  C_  ( F `  U ) )
 
Theoremmrcidb2 13392 A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( U  e.  C  <->  ( F `  U ) 
 C_  U ) )
 
Theoremmrcidm 13393 The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  ( F `  ( F `
  U ) )  =  ( F `  U ) )
 
Theoremmrcsscl 13394 The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  V  /\  V  e.  C )  ->  ( F `  U )  C_  V )
 
Theoremmrcuni 13395 Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  ~P X ) 
 ->  ( F `  U. U )  =  ( F ` 
 U. ( F " U ) ) )
 
Theoremmrcun 13396 Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  ( U  u.  V ) )  =  ( F `  (
 ( F `  U )  u.  ( F `  V ) ) ) )
 
Theoremsubmrc 13397 In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  F  =  (mrCls `  C )   &    |-  G  =  (mrCls `  ( C  i^i  ~P D ) )   =>    |-  ( ( C  e.  (Moore `  X )  /\  D  e.  C  /\  U  C_  D )  ->  ( G `  U )  =  ( F `  U ) )
 
7.2.2  Algebraic closure systems
 
Theoremisacs 13398* A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  E. f ( f : ~P X --> ~P X  /\  A. s  e.  ~P  X ( s  e.  C  <->  U. ( f "
 ( ~P s  i^i 
 Fin ) )  C_  s ) ) ) )
 
Theoremacsmre 13399 Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  ->  C  e.  (Moore `  X )
 )
 
Theoremisacs2 13400* In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( s  e.  C  <->  A. y  e.  ( ~P s  i^i  Fin )
 ( F `  y
 )  C_  s )
 ) )
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