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Theorem List for Metamath Proof Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-rest 13201* Function returning the subspace topology induced by the topology  y and the set  x. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
 |-t  =  ( j  e.  _V ,  x  e.  _V  |->  ran  (  y  e.  j  |->  ( y  i^i  x ) ) )
 
Definitiondf-topn 13202 Define the topology extractor function. This differs from df-tset 13101 when a structure has been restricted using df-ress 13029; in this case the TopSet component will still have a topology over the larger set, and this function fixes this by restricting the topology as well. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  TopOpen  =  ( w  e. 
 _V  |->  ( (TopSet `  w )t  ( Base `  w )
 ) )
 
Theoremrestfn 13203 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
 |-t  Fn  ( _V  X.  _V )
 
Theoremtopnfn 13204 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  TopOpen 
 Fn  _V
 
Theoremrestval 13205* The subspace topology induced by the topology  J on the set  A. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ran  (  x  e.  J  |->  ( x  i^i  A ) ) )
 
Theoremelrest 13206* The predicate "is an open set of a subspace topology". (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  ( ( J  e.  V  /\  B  e.  W )  ->  ( A  e.  ( Jt  B )  <->  E. x  e.  J  A  =  ( x  i^i  B ) ) )
 
Theoremelrestr 13207 Sufficient condition for being an open set in a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  ( ( J  e.  V  /\  S  e.  W  /\  A  e.  J ) 
 ->  ( A  i^i  S )  e.  ( Jt  S ) )
 
Theorem0rest 13208 Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( (/)t  A )  =  (/)
 
Theoremrestid2 13209 The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  J )
 
Theoremrestsspw 13210 The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( Jt  A )  C_  ~P A
 
Theoremfirest 13211 The finite intersections operator commutes with restriction. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( fi `  ( Jt  A ) )  =  ( ( fi `  J )t  A )
 
Theoremrestid 13212 The subspace topology of the base set is the original topology. (Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  V  ->  ( Jt  X )  =  J )
 
Theoremtopnval 13213 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (TopSet `  W )   =>    |-  ( Jt  B )  =  (
 TopOpen `  W )
 
Theoremtopnid 13214 Value of the topology extractor function when the topology is defined over the same set as the base. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (TopSet `  W )   =>    |-  ( J  C_  ~P B  ->  J  =  ( TopOpen `  W ) )
 
Theoremtopnpropd 13215 The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.)
 |-  ( ph  ->  ( Base `  K )  =  ( Base `  L )
 )   &    |-  ( ph  ->  (TopSet `  K )  =  (TopSet `  L ) )   =>    |-  ( ph  ->  (
 TopOpen `  K )  =  ( TopOpen `  L )
 )
 
Syntaxctg 13216 Extend class notation with a function that converts a basis to its corresponding topology.
 class  topGen
 
Syntaxcpt 13217 Extend class notation with a function whose value is a product topology.
 class  Xt_
 
Definitiondf-topgen 13218* Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78 (see tgval2 16526). See tgval3 16533 for an alternate expression for the value. (Contributed by NM, 16-Jul-2006.)
 |-  topGen  =  ( x  e. 
 _V  |->  { y  |  y 
 C_  U. ( x  i^i  ~P y ) } )
 
Definitiondf-pt 13219* Define the product topology on a collection of topologies. For convenience, it is defined on arbitrary collections of sets, expressed as a function from some index set to the subbases of each factor space. (Contributed by Mario Carneiro, 3-Feb-2015.)
 |- 
 Xt_  =  ( f  e.  _V  |->  ( topGen `  { x  |  E. g ( ( g  Fn  dom  f  /\  A. y  e.  dom  f ( g `  y )  e.  (
 f `  y )  /\  E. z  e.  Fin  A. y  e.  ( dom  f  \  z ) ( g `  y
 )  =  U. (
 f `  y )
 )  /\  x  =  X_ y  e.  dom  f
 ( g `  y
 ) ) } )
 )
 
Syntaxcprds 13220 The function constructing structure products.
 class  X_s
 
Syntaxcpws 13221 The function constructing structure powers.
 class  ^s
 
Definitiondf-prds 13222* Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  X_s  =  ( s  e.  _V ,  r  e.  _V  |->  [_ X_ x  e.  dom  r ( Base `  (
 r `  x )
 )  /  v ]_ [_ ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `
  x ) ( 
 Hom  `  ( r `  x ) ) ( g `  x ) ) )  /  h ]_ ( ( { <. (
 Base `  ndx ) ,  v >. ,  <. ( +g  ` 
 ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( +g  `  ( r `  x ) ) ( g `
  x ) ) ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `
  x ) ( .r `  ( r `
  x ) ) ( g `  x ) ) ) )
 >. }  u.  { <. (Scalar `  ndx ) ,  s >. ,  <. ( .s `  ndx ) ,  ( f  e.  ( Base `  s
 ) ,  g  e.  v  |->  ( x  e. 
 dom  r  |->  ( f ( .s `  (
 r `  x )
 ) ( g `  x ) ) ) ) >. } )  u.  ( { <. (TopSet `  ndx ) ,  ( Xt_ `  ( TopOpen  o.  r )
 ) >. ,  <. ( le ` 
 ndx ) ,  { <. f ,  g >.  |  ( { f ,  g }  C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  ( r `  x ) ) ( g `
  x ) ) } >. ,  <. ( dist ` 
 ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  sup ( ( ran  (  x  e.  dom  r  |->  ( ( f `  x ) ( dist `  (
 r `  x )
 ) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. (  Hom  `  ndx ) ,  h >. , 
 <. (comp `  ndx ) ,  ( a  e.  (
 v  X.  v ) ,  c  e.  v  |->  ( d  e.  (
 c h ( 2nd `  a ) ) ,  e  e.  ( h `
  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x ) ( <. ( ( 1st `  a
 ) `  x ) ,  ( ( 2nd `  a
 ) `  x ) >. (comp `  ( r `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. } )
 ) )
 
Theoremreldmprds 13223 The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |- 
 Rel  dom  X_s
 
Definitiondf-pws 13224* Define a structure power, which is just a structure product where all the factors are the same. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |- 
 ^s  =  ( r  e. 
 _V ,  i  e. 
 _V  |->  ( (Scalar `  r
 ) X_s ( i  X.  {
 r } ) ) )
 
Theoremprdsbasex 13225* Lemma for structure products. (Contributed by Mario Carneiro, 3-Jan-2015.)
 |-  B  =  X_ x  e.  dom  R ( Base `  ( R `  x ) )   =>    |-  B  e.  _V
 
Theoremimasvalstr 13226 Structure product value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
 <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )  u.  { <. (TopSet `  ndx ) ,  O >. ,  <. ( le `  ndx ) ,  L >. , 
 <. ( dist `  ndx ) ,  D >. } )   =>    |-  U Struct  <. 1 , ; 1
 2 >.
 
Theoremprdsvalstr 13227 Structure product value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )  u.  ( { <. (TopSet `  ndx ) ,  O >. ,  <. ( le ` 
 ndx ) ,  L >. ,  <. ( dist `  ndx ) ,  D >. }  u.  { <. (  Hom  ` 
 ndx ) ,  H >. ,  <. (comp `  ndx ) ,  .xb  >. } )
 ) Struct  <. 1 , ; 1 5 >.
 
Theoremprdsvallem 13228 Lemma for prdsbas 13231 and similar theorems. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( ph  ->  U  =  ( ( { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )  u.  ( { <. (TopSet `  ndx ) ,  O >. ,  <. ( le ` 
 ndx ) ,  L >. ,  <. ( dist `  ndx ) ,  D >. }  u.  { <. (  Hom  ` 
 ndx ) ,  H >. ,  <. (comp `  ndx ) ,  .xb  >. } )
 ) )   &    |-  A  =  ( E `  U )   &    |-  E  = Slot  ( E ` 
 ndx )   &    |-  ( ph  ->  T  e.  _V )   &    |-  { <. ( E `  ndx ) ,  T >. }  C_  (
 ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )  u.  ( { <. (TopSet `  ndx ) ,  O >. ,  <. ( le ` 
 ndx ) ,  L >. ,  <. ( dist `  ndx ) ,  D >. }  u.  { <. (  Hom  ` 
 ndx ) ,  H >. ,  <. (comp `  ndx ) ,  .xb  >. } )
 )   =>    |-  ( ph  ->  A  =  T )
 
Theoremprdsval 13229* Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.)
 |-  P  =  ( S
 X_s
 R )   &    |-  K  =  (
 Base `  S )   &    |-  ( ph  ->  dom  R  =  I )   &    |-  ( ph  ->  B  =  X_ x  e.  I  ( Base `  ( R `  x ) ) )   &    |-  ( ph  ->  .+  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( +g  `  ( R `  x ) ) ( g `  x ) ) ) ) )   &    |-  ( ph  ->  .X. 
 =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `
  x ) ( .r `  ( R `
  x ) ) ( g `  x ) ) ) ) )   &    |-  ( ph  ->  .x. 
 =  ( f  e.  K ,  g  e.  B  |->  ( x  e.  I  |->  ( f ( .s `  ( R `
  x ) ) ( g `  x ) ) ) ) )   &    |-  ( ph  ->  O  =  ( Xt_ `  ( TopOpen  o.  R ) ) )   &    |-  ( ph  ->  .<_  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x ) ) ( g `  x ) ) } )   &    |-  ( ph  ->  D  =  ( f  e.  B ,  g  e.  B  |->  sup (
 ( ran  (  x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x ) ) ( g `  x ) ) )  u.  {
 0 } ) , 
 RR* ,  <  ) ) )   &    |-  ( ph  ->  H  =  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( ( f `  x ) (  Hom  `  ( R `  x ) ) ( g `
  x ) ) ) )   &    |-  ( ph  ->  .xb 
 =  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a )
 ) ,  e  e.  ( H `  a
 )  |->  ( x  e.  I  |->  ( ( d `
  x ) (
 <. ( ( 1st `  a
 ) `  x ) ,  ( ( 2nd `  a
 ) `  x ) >. (comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  R  e.  Z )   =>    |-  ( ph  ->  P  =  ( ( { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. } )  u.  ( { <. (TopSet `  ndx ) ,  O >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. }  u.  { <. (  Hom  ` 
 ndx ) ,  H >. ,  <. (comp `  ndx ) ,  .xb  >. } )
 ) )
 
Theoremprdssca 13230 Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   =>    |-  ( ph  ->  S  =  (Scalar `  P )
 )
 
Theoremprdsbas 13231* Base set of a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   =>    |-  ( ph  ->  B  =  X_ x  e.  I  (
 Base `  ( R `  x ) ) )
 
Theoremprdsplusg 13232* Addition in a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   &    |-  .+  =  ( +g  `  P )   =>    |-  ( ph  ->  .+  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( +g  `  ( R `  x ) ) ( g `
  x ) ) ) ) )
 
Theoremprdsmulr 13233* Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   &    |-  .x. 
 =  ( .r `  P )   =>    |-  ( ph  ->  .x.  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( .r
 `  ( R `  x ) ) ( g `  x ) ) ) ) )
 
Theoremprdsvsca 13234* Scalar multiplication in a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   &    |-  K  =  ( Base `  S )   &    |-  .x.  =  ( .s `  P )   =>    |-  ( ph  ->  .x. 
 =  ( f  e.  K ,  g  e.  B  |->  ( x  e.  I  |->  ( f ( .s `  ( R `
  x ) ) ( g `  x ) ) ) ) )
 
Theoremprdsle 13235* Structure product weak ordering. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   &    |-  .<_  =  ( le `  P )   =>    |-  ( ph  ->  .<_  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x ) ) ( g `  x ) ) } )
 
Theoremprdsless 13236 Closure of the order relation on a structure product. (Contributed by Mario Carneiro, 16-Aug-2015.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   &    |-  .<_  =  ( le `  P )   =>    |-  ( ph  ->  .<_  C_  ( B  X.  B ) )
 
Theoremprdsds 13237* Structure product distance function. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   &    |-  D  =  ( dist `  P )   =>    |-  ( ph  ->  D  =  ( f  e.  B ,  g  e.  B  |->  sup ( ( ran  (  x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x ) ) ( g `  x ) ) )  u. 
 { 0 } ) ,  RR* ,  <  )
 ) )
 
Theoremprdsdsfn 13238 Structure product distance function. (Contributed by Mario Carneiro, 15-Sep-2015.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   &    |-  D  =  ( dist `  P )   =>    |-  ( ph  ->  D  Fn  ( B  X.  B ) )
 
Theoremprdstset 13239 Structure product topology. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   &    |-  O  =  (TopSet `  P )   =>    |-  ( ph  ->  O  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
 
Theoremprdshom 13240* Structure product hom-sets. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   &    |-  H  =  (  Hom  `  P )   =>    |-  ( ph  ->  H  =  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( ( f `  x ) (  Hom  `  ( R `  x ) ) ( g `  x ) ) ) )
 
Theoremprdsco 13241* Structure product composition operation. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   &    |-  H  =  (  Hom  `  P )   &    |-  .xb  =  (comp `  P )   =>    |-  ( ph  ->  .xb  =  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a
 ) ) ,  e  e.  ( H `  a
 )  |->  ( x  e.  I  |->  ( ( d `
  x ) (
 <. ( ( 1st `  a
 ) `  x ) ,  ( ( 2nd `  a
 ) `  x ) >. (comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) )
 
Theoremprdsbas2 13242* The base set of a structure product is an indexed set product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   =>    |-  ( ph  ->  B  =  X_ x  e.  I  ( Base `  ( R `  x ) ) )
 
Theoremprdsbasmpt 13243* A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   =>    |-  ( ph  ->  ( ( x  e.  I  |->  U )  e.  B  <->  A. x  e.  I  U  e.  ( Base `  ( R `  x ) ) ) )
 
Theoremprdsbasfn 13244 Points in the structure product are functions; use this with dffn5 5420 to establish equalities. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  ( ph  ->  T  e.  B )   =>    |-  ( ph  ->  T  Fn  I )
 
Theoremprdsbasprj 13245 Each point in a structure product restricts on each coordinate to the relevant base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  ( ph  ->  T  e.  B )   &    |-  ( ph  ->  J  e.  I )   =>    |-  ( ph  ->  ( T `  J )  e.  ( Base `  ( R `  J ) ) )
 
Theoremprdsplusgval 13246* Value of a componentwise sum in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  .+  =  ( +g  `  Y )   =>    |-  ( ph  ->  ( F  .+  G )  =  ( x  e.  I  |->  ( ( F `
  x ) (
 +g  `  ( R `  x ) ) ( G `  x ) ) ) )
 
Theoremprdsplusgfval 13247 Value of a structure product sum at a single coordinate. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  .+  =  ( +g  `  Y )   &    |-  ( ph  ->  J  e.  I
 )   =>    |-  ( ph  ->  (
 ( F  .+  G ) `  J )  =  ( ( F `  J ) ( +g  `  ( R `  J ) ) ( G `
  J ) ) )
 
Theoremprdsmulrval 13248* Value of a componentwise ring product in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  .x.  =  ( .r `  Y )   =>    |-  ( ph  ->  ( F  .x.  G )  =  ( x  e.  I  |->  ( ( F `  x ) ( .r
 `  ( R `  x ) ) ( G `  x ) ) ) )
 
Theoremprdsmulrfval 13249 Value of a structure product's ring product at a single coordinate. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  .x.  =  ( .r `  Y )   &    |-  ( ph  ->  J  e.  I
 )   =>    |-  ( ph  ->  (
 ( F  .x.  G ) `  J )  =  ( ( F `  J ) ( .r
 `  ( R `  J ) ) ( G `  J ) ) )
 
Theoremprdsleval 13250* Value of the product ordering in a structure product. (Contributed by Mario Carneiro, 15-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  .<_  =  ( le `  Y )   =>    |-  ( ph  ->  ( F  .<_  G  <->  A. x  e.  I  ( F `  x ) ( le `  ( R `  x ) ) ( G `  x ) ) )
 
Theoremprdsdsval 13251* Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  D  =  (
 dist `  Y )   =>    |-  ( ph  ->  ( F D G )  =  sup ( ( ran  (  x  e.  I  |->  ( ( F `
  x ) (
 dist `  ( R `  x ) ) ( G `  x ) ) )  u.  {
 0 } ) , 
 RR* ,  <  ) )
 
Theoremprdsvscaval 13252* Scalar multiplication in a structure product is pointwise. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .s `  Y )   &    |-  K  =  ( Base `  S )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  ( ph  ->  F  e.  K )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F  .x.  G )  =  ( x  e.  I  |->  ( F ( .s
 `  ( R `  x ) ) ( G `  x ) ) ) )
 
Theoremprdsvscafval 13253 Scalar multiplication of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .s `  Y )   &    |-  K  =  ( Base `  S )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  Fn  I )   &    |-  ( ph  ->  F  e.  K )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  J  e.  I )   =>    |-  ( ph  ->  (
 ( F  .x.  G ) `  J )  =  ( F ( .s
 `  ( R `  J ) ) ( G `  J ) ) )
 
Theoremprdsbas3 13254* The base set of an indexed structure product. (Contributed by Mario Carneiro, 13-Sep-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  A. x  e.  I  R  e.  X )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ph  ->  B  =  X_ x  e.  I  K )
 
Theoremprdsbasmpt2 13255* A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Mario Carneiro, 3-Jul-2015.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  A. x  e.  I  R  e.  X )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ph  ->  ( ( x  e.  I  |->  U )  e.  B  <->  A. x  e.  I  U  e.  K ) )
 
Theoremprdsbascl 13256* An element of the base has projections closed in the factors. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  A. x  e.  I  R  e.  X )   &    |-  K  =  (
 Base `  R )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  A. x  e.  I  ( F `  x )  e.  K )
 
Theoremprdsdsval2 13257* Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  A. x  e.  I  R  e.  X )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  E  =  ( dist `  R )   &    |-  D  =  (
 dist `  Y )   =>    |-  ( ph  ->  ( F D G )  =  sup ( ( ran  (  x  e.  I  |->  ( ( F `
  x ) E ( G `  x ) ) )  u. 
 { 0 } ) ,  RR* ,  <  )
 )
 
Theoremprdsdsval3 13258* Value of the metric in a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  A. x  e.  I  R  e.  X )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  K  =  ( Base `  R )   &    |-  E  =  ( ( dist `  R )  |`  ( K  X.  K ) )   &    |-  D  =  (
 dist `  Y )   =>    |-  ( ph  ->  ( F D G )  =  sup ( ( ran  (  x  e.  I  |->  ( ( F `
  x ) E ( G `  x ) ) )  u. 
 { 0 } ) ,  RR* ,  <  )
 )
 
Theorempwsval 13259 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 13260 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 13261 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 13262 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 13263 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 13264 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 13265 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 13266* 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 13267 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 13268 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 13269 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 13270 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 13271* 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 13272 Extend class notation with the order topology.
 class ordTop
 
Syntaxcxrs 13273 Extend class notation with the extended real number structure.
 class  RR*
 s
 
Syntaxc0g 13274 Extend class notation with group identity element.
 class  0g
 
Syntaxcgsu 13275 Extend class notation to include finitely supported group sums.
 class  gsumg
 
Definitiondf-ordt 13276* 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 13277* The extended real number structure. Unlike df-cnfld 16210, 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 16210. 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 13278* 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 13279* 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 17641. (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 13280 Extend class notation with the quotient topology function.
 class qTop
 
Syntaxcimas 13281 Image structure function.
 class  "s
 
Syntaxcqus 13282 Quotient structure function.
 class  /.s
 
Syntaxcxps 13283 Binary product structure function.
 class  X.s
 
Definitiondf-qtop 13284* 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 13285* 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 4601, 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 4600) and/or  R ( with df-ress 13029) 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 13286* Define a quotient ring (or quotient group), which is a special case of an image structure df-imas 13285 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 13287* 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 13288* 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 13289 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 13290* 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 13291 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 13292* 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 13293* 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 13294* 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 13295* 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 13296 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 13297* 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 13298 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 13299 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 13300 Lemma for f1ocpbl 13301. (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 ) ) )
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