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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempwsval 13401 Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  F  =  (Scalar `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  W )  ->  Y  =  ( F X_s ( I  X.  { R } ) ) )
 
Theorempwsbas 13402 Base set of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( B  ^m  I )  =  ( Base `  Y )
 )
 
Theorempwselbasb 13403 Membership in the base set of a structure product. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  R )   &    |-  V  =  ( Base `  Y )   =>    |-  (
 ( R  e.  W  /\  I  e.  Z )  ->  ( X  e.  V 
 <->  X : I --> B ) )
 
Theorempwselbas 13404 An element of a structure power is a function from the index set to the base set of the structure. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  R )   &    |-  V  =  ( Base `  Y )   &    |-  ( ph  ->  R  e.  W )   &    |-  ( ph  ->  I  e.  Z )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  X : I --> B )
 
Theorempwsplusgval 13405 Value of addition in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  Y )   =>    |-  ( ph  ->  ( F  .+b  G )  =  ( F  o F  .+  G ) )
 
Theorempwsmulrval 13406 Value of multiplication in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |- 
 .x.  =  ( .r `  R )   &    |-  .xb  =  ( .r `  Y )   =>    |-  ( ph  ->  ( F  .xb  G )  =  ( F  o F  .x.  G ) )
 
Theorempwsle 13407 Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  O  =  ( le `  R )   &    |- 
 .<_  =  ( le `  Y )   =>    |-  ( ( R  e.  V  /\  I  e.  W )  ->  .<_  =  (  o R O  i^i  ( B  X.  B ) ) )
 
Theorempwsleval 13408* Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  O  =  ( le `  R )   &    |- 
 .<_  =  ( le `  Y )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F  .<_  G  <->  A. x  e.  I  ( F `  x ) O ( G `  x ) ) )
 
Theorempwsvscafval 13409 Scalar multiplication in a structure power is pointwise. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .s `  R )   &    |-  .xb 
 =  ( .s `  Y )   &    |-  F  =  (Scalar `  R )   &    |-  K  =  (
 Base `  F )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( A  .xb  X )  =  ( ( I  X.  { A } )  o F  .x.  X )
 )
 
Theorempwsvscaval 13410 Scalar multiplication of a single coordinate in a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .s `  R )   &    |-  .xb 
 =  ( .s `  Y )   &    |-  F  =  (Scalar `  R )   &    |-  K  =  (
 Base `  F )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  J  e.  I )   =>    |-  ( ph  ->  (
 ( A  .xb  X ) `
  J )  =  ( A  .x.  ( X `  J ) ) )
 
Theorempwssca 13411 The ring of scalars of a structure product. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  S  =  (Scalar `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  W )  ->  S  =  (Scalar `  Y ) )
 
Theorempwsdiagel 13412 Membership of diagonal elements in the structure power base set. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  R )   &    |-  C  =  ( Base `  Y )   =>    |-  (
 ( ( R  e.  V  /\  I  e.  W )  /\  A  e.  B )  ->  ( I  X.  { A } )  e.  C )
 
Theorempwssnf1o 13413* Triviality of singleton powers: set equipollence. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( R 
 ^s 
 { I } )   &    |-  B  =  ( Base `  R )   &    |-  F  =  ( x  e.  B  |->  ( { I }  X.  { x } ) )   &    |-  C  =  ( Base `  Y )   =>    |-  ( ( R  e.  V  /\  I  e.  W )  ->  F : B -1-1-onto-> C )
 
7.1.4  Definition of the structure quotient
 
Syntaxcordt 13414 Extend class notation with the order topology.
 class ordTop
 
Syntaxcxrs 13415 Extend class notation with the extended real number structure.
 class  RR*
 s
 
Syntaxc0g 13416 Extend class notation with group identity element.
 class  0g
 
Syntaxcgsu 13417 Extend class notation to include finitely supported group sums.
 class  gsumg
 
Definitiondf-ordt 13418* 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 13419* The extended real number structure. Unlike df-cnfld 16394, 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 16394. 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 13420* 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 13421* 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 17825. (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 13422 Extend class notation with the quotient topology function.
 class qTop
 
Syntaxcimas 13423 Image structure function.
 class  "s
 
Syntaxcqus 13424 Quotient structure function.
 class  /.s
 
Syntaxcxps 13425 Binary product structure function.
 class  X.s
 
Definitiondf-qtop 13426* 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 13427* 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 4718, 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 4717) and/or  R ( with df-ress 13171) 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 13428* Define a quotient ring (or quotient group), which is a special case of an image structure df-imas 13427 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 13429* 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 13430* 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 13431 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 13432* 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 13433 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 13434* 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 13435* 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 13436* 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 13437* 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 13438 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 13439* 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 13440 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 13441 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 13442 Lemma for f1ocpbl 13443. (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 13443 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 13444 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 13445 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 13446* 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 13447* 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 13448* 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 13449* 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 13450* 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 13451* 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 13452* 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 13453* 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 13454* 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 13455* 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 13456* 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 13457* 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 13458 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 13459* 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 13460* 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 13461* The function in divsval 13460 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 13462 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 13463 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 13464 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 13465* Value of the function in divsval 13460. (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 13466* Lemma for ercpbl 13467. (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 13467* 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 13468* 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 13469* 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 13470* 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 13471* 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 13472* 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 13473* 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 13474* 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 13475 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 13476 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 13477 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 13478 The pair function maps  0 to  A. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( A  e.  V  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
 
Theoremxpsc1 13479 The pair function maps  1 to  B. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B } ) `  1o )  =  B )
 
Theoremxpscfv 13480 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 13481* Elementhood in the target space of the function  F appearing in xpsval 13490. (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 13482 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 13483* Elementhood in the target space of the function  F appearing in xpsval 13490. (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 13484 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 13485* The value of the function appearing in xpsval 13490. (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 13486* The function appearing in xpsval 13490 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 13487* 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 13488* 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 13489* The function appearing in xpsval 13490 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 13490* 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 13491* The indexed structure product that appears in xpsval 13490 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 13492 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 13493* Lemma for xpsadd 13494 and xpsmul 13495. (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 13494 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 13495 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 13496 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 13497 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 13498 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 13499 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 13500 The class of Moore systems.
 class Moore
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