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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcgsu 13401 Extend class notation to include finitely supported group sums.
 class  gsumg
 
Definitiondf-ordt 13402* Define the order topology, given an order  <_, written as  r below. A closed subbasis for the order topology is given by the closed rays  [ y , 
+oo )  =  {
z  e.  X  | 
y  <_  z } and  (  -oo , 
y ]  =  {
z  e.  X  | 
z  <_  y }, along with  ( 
-oo ,  +oo )  =  X itself. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |- ordTop  =  ( r  e.  _V  |->  ( topGen `  ( fi `  ( { dom  r }  u.  ran  ( ( x  e.  dom  r  |->  { y  e.  dom  r  |  -.  y r x } )  u.  ( x  e.  dom  r  |->  { y  e.  dom  r  |  -.  x r y } ) ) ) ) ) )
 
Definitiondf-xrs 13403* The extended real number structure. Unlike df-cnfld 16378, the extended real numbers do not have good algebraic properties, so this is not actually a group or anything higher, even though it has just as many operations as df-cnfld 16378. The main interest in this structure is in its ordering, which is complete and compact. The metric described here is an extension of the absolute value metric, but it is not itself a metric because  +oo is infinitely far from all other points. The topology is based on the order and not the extended metric (which would make  +oo an isolated point since there is nothing else in the  1 -ball around it). All components of this structure agree with ℂfld when restricted to  RR. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  RR* s  =  ( { <. ( Base `  ndx ) , 
 RR* >. ,  <. ( +g  ` 
 ndx ) ,  + e >. ,  <. ( .r
 `  ndx ) ,  x e >. }  u.  { <. (TopSet `  ndx ) ,  (ordTop `  <_  ) >. , 
 <. ( le `  ndx ) ,  <_  >. ,  <. (
 dist `  ndx ) ,  ( x  e.  RR* ,  y  e.  RR*  |->  if ( x  <_  y ,  (
 y + e  - e x ) ,  ( x + e  - e
 y ) ) )
 >. } )
 
Definitiondf-0g 13404* Define group identity element. (Contributed by NM, 20-Aug-2011.)
 |- 
 0g  =  ( g  e.  _V  |->  ( iota
 e ( e  e.  ( Base `  g )  /\  A. x  e.  ( Base `  g ) ( ( e ( +g  `  g ) x )  =  x  /\  ( x ( +g  `  g
 ) e )  =  x ) ) ) )
 
Definitiondf-gsum 13405* Define the group sum for the structure  G of a finite sequence of elements whose values are defined by the expression  B and whose set of indices is  A. It may be viewed as a product (if 
G is a multiplication), a sum (if 
G is an addition) or whatever. The variable  k is normally a free variable in  B ( i.e.  B can be thought of as  B ( k )). The definition is meaningful in three contexts, depending on the size of the index set  A and each demanding different properties of  G.

1. If  A  =  (/) and  G has an identity element, then the sum equals this identity.

2. If  A  =  ( M ... N ) and 
G is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e.  ( B ( 1 )  +  B
( 2 ) )  +  B ( 3 ) etc.

3. If  A is a finite set (or is non-zero for finitely many indices) and  G is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined.

4. If  A is an infinite set and  G is a Hausdorff topological group, then there is a meaningful sum, but  gsumg cannot handle this case. See df-tsms 17809. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.)

 |- 
 gsumg  =  ( w  e.  _V ,  f  e.  _V  |->  [_
 { x  e.  ( Base `  w )  | 
 A. y  e.  ( Base `  w ) ( ( x ( +g  `  w ) y )  =  y  /\  (
 y ( +g  `  w ) x )  =  y ) }  /  o ]_ if ( ran  f  C_  o ,  ( 0g
 `  w ) ,  if ( dom  f  e.  ran  ... ,  ( iota
 x E. m E. n  e.  ( ZZ>= `  m ) ( dom  f  =  ( m
 ... n )  /\  x  =  (  seq  m ( ( +g  `  w ) ,  f ) `  n ) ) ) ,  ( iota x E. g [. ( `' f " ( _V  \  o ) )  /  y ]. ( g : ( 1 ... ( # `
  y ) ) -1-1-onto-> y 
 /\  x  =  ( 
 seq  1 ( (
 +g  `  w ) ,  ( f  o.  g
 ) ) `  ( # `
  y ) ) ) ) ) ) )
 
Syntaxcqtop 13406 Extend class notation with the quotient topology function.
 class qTop
 
Syntaxcimas 13407 Image structure function.
 class  "s
 
Syntaxcqus 13408 Quotient structure function.
 class  /.s
 
Syntaxcxps 13409 Binary product structure function.
 class  X.s
 
Definitiondf-qtop 13410* Define the quotient topology given a function  f and topology  j on the domain of  f. (Contributed by Mario Carneiro, 23-Mar-2015.)
 |- qTop  =  ( j  e.  _V ,  f  e.  _V  |->  { s  e.  ~P (
 f " U. j )  |  ( ( `' f " s )  i^i  U. j )  e.  j } )
 
Definitiondf-imas 13411* Define an image structure, which takes a structure and a function on the base set, and maps all the operations via the function. For this to work properly  f must either be injective or satisfy the well-definedness condition  f ( a )  =  f ( c )  /\  f ( b )  =  f ( d )  ->  f (
a  +  b )  =  f ( c  +  d ) for each relevant operation.

Note that although we call this an "image" by association to df-ima 4702, in order to keep the definition simple we consider only the case when the domain of  F is equal to the base set of  R. Other cases can be achieved by restricting 
F (with df-res 4701) and/or  R ( with df-ress 13155) to their common domain. (Contributed by Mario Carneiro, 23-Feb-2015.)

 |-  "s  =  ( f  e.  _V ,  r  e.  _V  |->  [_ ( Base `  r )  /  v ]_ ( ( { <. ( Base `  ndx ) ,  ran  f >. , 
 <. ( +g  `  ndx ) ,  U_ p  e.  v  U_ q  e.  v  { <. <. ( f `
  p ) ,  ( f `  q
 ) >. ,  ( f `
  ( p (
 +g  `  r )
 q ) ) >. }
 >. ,  <. ( .r `  ndx ) ,  U_ p  e.  v  U_ q  e.  v  { <. <. ( f `
  p ) ,  ( f `  q
 ) >. ,  ( f `
  ( p ( .r `  r ) q ) ) >. }
 >. }  u.  { <. (Scalar `  ndx ) ,  (Scalar `  r ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  v  ( p  e.  ( Base `  (Scalar `  r
 ) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s `  r
 ) q ) ) ) >. } )  u. 
 { <. (TopSet `  ndx ) ,  ( ( TopOpen `  r
 ) qTop  f ) >. , 
 <. ( le `  ndx ) ,  ( (
 f  o.  ( le `  r ) )  o.  `' f ) >. ,  <. (
 dist `  ndx ) ,  ( x  e.  ran  f ,  y  e.  ran  f  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  (
 ( v  X.  v
 )  ^m  ( 1 ... n ) )  |  ( ( f `  ( 1st `  ( h `  1 ) ) )  =  x  /\  (
 f `  ( 2nd `  ( h `  n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  -  1 ) ) ( f `  ( 2nd `  ( h `  i ) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s 
 gsumg  ( ( dist `  r
 )  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. } ) )
 
Definitiondf-divs 13412* Define a quotient ring (or quotient group), which is a special case of an image structure df-imas 13411 where the image function is  x  |->  [ x ] e. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |- 
 /.s 
 =  ( r  e. 
 _V ,  e  e. 
 _V  |->  ( ( x  e.  ( Base `  r
 )  |->  [ x ] e
 )  "s  r ) )
 
Definitiondf-xps 13413* Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 X.s 
 =  ( r  e. 
 _V ,  s  e. 
 _V  |->  ( `' ( x  e.  ( Base `  r ) ,  y  e.  ( Base `  s )  |->  `' ( { x }  +c  { y } )
 )  "s  ( (Scalar `  r
 ) X_s `' ( { r }  +c  { s } )
 ) ) )
 
Theoremimasval 13414* Value of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  G  =  (Scalar `  R )   &    |-  K  =  ( Base `  G )   &    |-  .x.  =  ( .s `  R )   &    |-  J  =  ( TopOpen `  R )   &    |-  E  =  (
 dist `  R )   &    |-  N  =  ( le `  R )   &    |-  ( ph  ->  .+b  =  U_ p  e.  V  U_ q  e.  V  { <. <.
 ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p  .+  q ) ) >. } )   &    |-  ( ph  ->  .xb 
 =  U_ p  e.  V  U_ q  e.  V  { <.
 <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p  .X.  q ) ) >. } )   &    |-  ( ph  ->  .(x)  = 
 U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  O  =  ( J qTop  F ) )   &    |-  ( ph  ->  D  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  (
 ( V  X.  V )  ^m  ( 1 ... n ) )  |  ( ( F `  ( 1st `  ( h `  1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( h `  n ) ) )  =  y  /\  A. i  e.  ( 1
 ... ( n  -  1 ) ) ( F `  ( 2nd `  ( h `  i
 ) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1
 ) ) ) ) ) }  |->  ( RR* s 
 gsumg  ( E  o.  g
 ) ) ) , 
 RR* ,  `'  <  ) ) )   &    |-  ( ph  ->  .<_  =  ( ( F  o.  N )  o.  `' F ) )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   =>    |-  ( ph  ->  U  =  ( ( { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .xb  >. }  u.  {
 <. (Scalar `  ndx ) ,  G >. ,  <. ( .s
 `  ndx ) ,  .(x)  >. } )  u.  { <. (TopSet `  ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
 dist `  ndx ) ,  D >. } ) )
 
Theoremimasbas 13415 The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   =>    |-  ( ph  ->  B  =  ( Base `  U ) )
 
Theoremimasds 13416* The distance function of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( dist `  R )   &    |-  D  =  ( dist `  U )   =>    |-  ( ph  ->  D  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  (
 ( F `  ( 1st `  ( h `  1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( h `  n ) ) )  =  y  /\  A. i  e.  ( 1
 ... ( n  -  1 ) ) ( F `  ( 2nd `  ( h `  i
 ) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1
 ) ) ) ) ) }  |->  ( RR* s 
 gsumg  ( E  o.  g
 ) ) ) , 
 RR* ,  `'  <  ) ) )
 
Theoremimasdsfn 13417 The distance function is a function on the base set. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( dist `  R )   &    |-  D  =  ( dist `  U )   =>    |-  ( ph  ->  D  Fn  ( B  X.  B ) )
 
Theoremimasdsval 13418* The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( dist `  R )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  S  =  { h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  (
 ( F `  ( 1st `  ( h `  1 ) ) )  =  X  /\  ( F `  ( 2nd `  ( h `  n ) ) )  =  Y  /\  A. i  e.  ( 1
 ... ( n  -  1 ) ) ( F `  ( 2nd `  ( h `  i
 ) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1
 ) ) ) ) ) }   =>    |-  ( ph  ->  ( X D Y )  = 
 sup ( U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR* s  gsumg  ( E  o.  g ) ) ) ,  RR* ,  `'  <  ) )
 
Theoremimasdsval2 13419* The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  E  =  ( dist `  R )   &    |-  D  =  ( dist `  U )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  S  =  { h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  (
 ( F `  ( 1st `  ( h `  1 ) ) )  =  X  /\  ( F `  ( 2nd `  ( h `  n ) ) )  =  Y  /\  A. i  e.  ( 1
 ... ( n  -  1 ) ) ( F `  ( 2nd `  ( h `  i
 ) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1
 ) ) ) ) ) }   &    |-  T  =  ( E  |`  ( V  X.  V ) )   =>    |-  ( ph  ->  ( X D Y )  =  sup ( U_ n  e.  NN  ran  (
 g  e.  S  |->  (
 RR* s  gsumg  ( T  o.  g
 ) ) ) , 
 RR* ,  `'  <  ) )
 
Theoremimasplusg 13420* The group operation in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  U )   =>    |-  ( ph  ->  .+b  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `
  p ) ,  ( F `  q
 ) >. ,  ( F `
  ( p  .+  q ) ) >. } )
 
Theoremimasmulr 13421* The ring multiplication in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  U )   =>    |-  ( ph  ->  .xb  =  U_ p  e.  V  U_ q  e.  V  { <. <.
 ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p  .x.  q ) ) >. } )
 
Theoremimassca 13422 The scalar field of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  G  =  (Scalar `  R )   =>    |-  ( ph  ->  G  =  (Scalar `  U ) )
 
Theoremimasvsca 13423* The scalar multiplication operation of an image structure. (Contributed by Mario Carneiro, 23-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  ->  .xb  =  U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) )
 
Theoremimastset 13424 The topology of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  J  =  ( TopOpen `  R )   &    |-  O  =  (TopSet `  U )   =>    |-  ( ph  ->  O  =  ( J qTop  F ) )
 
Theoremimasle 13425 The ordering of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ph  ->  R  e.  Z )   &    |-  N  =  ( le `  R )   &    |- 
 .<_  =  ( le `  U )   =>    |-  ( ph  ->  .<_  =  ( ( F  o.  N )  o.  `' F ) )
 
Theoremf1ocpbllem 13426 Lemma for f1ocpbl 13427. (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  =  C  /\  B  =  D ) ) )
 
Theoremf1ocpbl 13427 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 13428 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 13429 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 13430* 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 13431* 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 13432* 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 13433* 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 13434* 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 13435* 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 13436* 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 13437* 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 13438* 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 13439* 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 13440* 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 13441* 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 13442 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 13443* 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 13444* 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 13445* The function in divsval 13444 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 13446 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 13447 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 13448 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 13449* Value of the function in divsval 13444. (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 13450* Lemma for ercpbl 13451. (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 13451* 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 13452* 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 13453* 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 13454* 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 13455* 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 13456* 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 13457* 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 13458* 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 13459 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 13460 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 13461 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 13462 The pair function maps  0 to  A. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( A  e.  V  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
 
Theoremxpsc1 13463 The pair function maps  1 to  B. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B } ) `  1o )  =  B )
 
Theoremxpscfv 13464 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 13465* Elementhood in the target space of the function  F appearing in xpsval 13474. (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 13466 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 13467* Elementhood in the target space of the function  F appearing in xpsval 13474. (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 13468 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 13469* The value of the function appearing in xpsval 13474. (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 13470* The function appearing in xpsval 13474 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 13471* 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 13472* 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 13473* The function appearing in xpsval 13474 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 13474* 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 13475* The indexed structure product that appears in xpsval 13474 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 13476 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 13477* Lemma for xpsadd 13478 and xpsmul 13479. (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 13478 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 13479 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 13480 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 13481 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 13482 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 13483 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 13484 The class of Moore systems.
 class Moore
 
Syntaxcmrc 13485 The class function generating Moore closures.
 class mrCls
 
Syntaxcmri 13486 mrInd is a class function which takes a Moore system to its set of independent sets.
 class mrInd
 
Syntaxcacs 13487 The class of algebraic closure (Moore) systems.
 class ACS
 
Definitiondf-mre 13488* 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 16815) and vector spaces (lssmre 15723) 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 13492, mresspw 13494, mre1cl 13496 and mreintcl 13497 for the major properties of a Moore collection. Note that a Moore collection uniquely determines its base set (mreuni 13502); as such the disjoint union of all Moore collections is sometimes considered as  U. ran Moore, justified by mreunirn 13503. (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 13489* 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 16816) and linear span (mrclsp 15746).

A Moore closure operation  N is (1) extensive, i.e.,  x  C_  ( N `  x ) for all subsets  x of the base set (mrcssid 13519), (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 13518), and (3) idempotent, i.e.,  ( N `  ( N `  x )
)  =  ( N `
 x ) for all subsets  x of the base set (mrcidm 13521.) 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 13490* 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 13491* 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 7344 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 13492* 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 13493 The Moore collection generator is a well-behaved function. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |- Moore  Fn  _V
 
Theoremmresspw 13494 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 13495 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 13496 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 13497 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 13498* 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 13499 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 13500* 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 )
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