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Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremimasaddvallem 14801* 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 14802* 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 14803* 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 14804* 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 14805* 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 14806* 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 14807* 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 14808* 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 14809* 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 14810* 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 14811* 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 14812 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 14813* 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 ) )
 
Theoremqusval 14814* 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 )
 )
 
Theoremquslem 14815* The function in qusval 14814 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 /.  .~  ) )
 
Theoremqusin 14816 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 ) ) ) )
 
Theoremqusbas 14817 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 )
 )
 
Theoremquss 14818 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 14819* Value of the function in qusval 14814. (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 14820* Lemma for ercpbl 14821. (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 14821* Translate the function compatibility 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 14822* Translate the relation compatibility 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 ) ) )
 
Theoremqusaddvallem 14823* 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 ) ]  .~  )
 
Theoremqusaddflem 14824* 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
 /.  .~  ) )
 
Theoremqusaddval 14825* 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 ) ]  .~  )
 
Theoremqusaddf 14826* 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
 /.  .~  ) )
 
Theoremqusmulval 14827* 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 ) ]  .~  )
 
Theoremqusmulf 14828* 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 14829 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 14830 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 14831 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 14832 The pair function maps  0 to  A. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( A  e.  V  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
 
Theoremxpsc1 14833 The pair function maps  1 to  B. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B } ) `  1o )  =  B )
 
Theoremxpscfv 14834 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 14835* Elementhood in the target space of the function  F appearing in xpsval 14844. (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 14836 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 14837* Elementhood in the target space of the function  F appearing in xpsval 14844. (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 14838 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 14839* The value of the function appearing in xpsval 14844. (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 14840* The function appearing in xpsval 14844 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 14841* 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 14842* 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 14843* The function appearing in xpsval 14844 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 14844* 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 14845* The indexed structure product that appears in xpsval 14844 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 14846 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 14847* Lemma for xpsadd 14848 and xpsmul 14849. (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 14848 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 14849 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 14850 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 14851 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 14852 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 14853 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 14854 The class of Moore systems.
 class Moore
 
Syntaxcmrc 14855 The class function generating Moore closures.
 class mrCls
 
Syntaxcmri 14856 mrInd is a class function which takes a Moore system to its set of independent sets.
 class mrInd
 
Syntaxcacs 14857 The class of algebraic closure (Moore) systems.
 class ACS
 
Definitiondf-mre 14858* 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 19447) and vector spaces (lssmre 17483) to the most general setting in which such concepts make sense. Definition of Moore collection of sets in [Schechter] p. 78. A Moore collection may also be called a closure system (Section 0.6 in [Gratzer] p. 23.) The name Moore collection is after Eliakim Hastings Moore, who discussed these systems in Part I of [Moore] p. 53 to 76.

See ismre 14862, mresspw 14864, mre1cl 14866 and mreintcl 14867 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 14872); as such the disjoint union of all Moore collections is sometimes considered as  U. ran Moore, justified by mreunirn 14873. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by David Moews, 1-May-2017.)

 |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }
 )
 
Definitiondf-mrc 14859* 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 19448) and linear span (mrclsp 17506).

A Moore closure operation  N is (1) extensive, i.e.,  x  C_  ( N `  x ) for all subsets  x of the base set (mrcssid 14889), (2) isotone, i.e.,  x  C_  y implies that  ( N `
 x )  C_  ( N `  y ) for all subsets  x and  y of the base set (mrcss 14888), and (3) idempotent, i.e.,  ( N `  ( N `  x )
)  =  ( N `
 x ) for all subsets  x of the base set (mrcidm 14891.) Operators satisfying these three properties are in bijective correspondence with Moore collections, so these properties may be used to give an alternate characterization of a Moore collection by providing a closure operation  N on the set of subsets of a given base set which satisfies (1), (2), and (3); the closed sets can be recovered as those sets which equal their closures (Section 4.5 in [Schechter] p. 82.) (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by David Moews, 1-May-2017.)

 |- mrCls  =  ( c  e.  U. ran Moore 
 |->  ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) )
 
Definitiondf-mri 14860* In a Moore system, a set is independent if no element of the set is in the closure of the set with the element removed (Section 0.6 in [Gratzer] p. 27; Definition 4.1.1 in [FaureFrolicher] p. 83.) mrInd is a class function which takes a Moore system to its set of independent sets. (Contributed by David Moews, 1-May-2017.)
 |- mrInd  =  ( c  e.  U. ran Moore 
 |->  { s  e.  ~P U. c  |  A. x  e.  s  -.  x  e.  ( (mrCls `  c
 ) `  ( s  \  { x } )
 ) } )
 
Definitiondf-acs 14861* 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 8072 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 14862* 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 14863 The Moore collection generator is a well-behaved function. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |- Moore  Fn  _V
 
Theoremmresspw 14864 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 14865 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 14866 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 14867 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 14868* 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 14869 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 14870* 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 14871 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 14872 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 14873 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 14874* 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 14875* 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 14876 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 14877 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 14878* 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 14879 Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- mrCls  Fn  U. ran Moore
 
Theoremmrcfval 14880* 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 14881 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 14882* 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 14883 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 14884 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 14885 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 14886 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 14887 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 14888 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 14889 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 14890 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 14891 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 14892 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 14893 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 14894 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 ) ) ) )
 
Theoremmrcssvd 14895 The Moore closure of a set is a subset of the base. Deduction form of mrcssv 14886. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   =>    |-  ( ph  ->  ( N `  B )  C_  X )
 
Theoremmrcssd 14896 Moore closure preserves subset ordering. Deduction form of mrcss 14888. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  V )   &    |-  ( ph  ->  V  C_  X )   =>    |-  ( ph  ->  ( N `  U )  C_  ( N `  V ) )
 
Theoremmrcssidd 14897 A set is contained in its Moore closure. Deduction form of mrcssid 14889. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  X )   =>    |-  ( ph  ->  U 
 C_  ( N `  U ) )
 
Theoremmrcidmd 14898 Moore closure is idempotent. Deduction form of mrcidm 14891. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  U 
 C_  X )   =>    |-  ( ph  ->  ( N `  ( N `
  U ) )  =  ( N `  U ) )
 
Theoremmressmrcd 14899 In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (Moore `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  ( N `  T ) )   &    |-  ( ph  ->  T  C_  S )   =>    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )
 
Theoremsubmrc 14900 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 ) )
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