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Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremotpsle 13301 The order of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  B >. ,  <. (TopSet `  ndx ) ,  J >. , 
 <. ( le `  ndx ) ,  .<_  >. }   =>    |-  (  .<_  e.  V  -> 
 .<_  =  ( le `  K ) )
 
Theoremressle 13302  le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
 |-  W  =  ( Ks  A )   &    |-  .<_  =  ( le `  K )   =>    |-  ( A  e.  V  -> 
 .<_  =  ( le `  W ) )
 
Theoremocndx 13303 Index value of the df-ocomp 13225 slot. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  ( oc `  ndx )  = ; 1 1
 
Theoremocid 13304 Utility theorem: index-independent form of df-ocomp 13225. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |- 
 oc  = Slot  ( oc ` 
 ndx )
 
Theoremdsndx 13305 Index value of the df-ds 13226 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( dist `  ndx )  = ; 1
 2
 
Theoremdsid 13306 Utility theorem: index-independent form of df-ds 13226. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |- 
 dist  = Slot  ( dist `  ndx )
 
Theoremodrngstr 13307 Functionality of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  W Struct  <. 1 , ; 1 2 >.
 
Theoremodrngbas 13308 The base set of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  ( B  e.  V  ->  B  =  ( Base `  W ) )
 
Theoremodrngplusg 13309 The addition operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  (  .+  e.  V  ->  .+  =  ( +g  `  W ) )
 
Theoremodrngmulr 13310 The multiplication operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  (  .x.  e.  V  ->  .x.  =  ( .r
 `  W ) )
 
Theoremodrngtset 13311 The open sets of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  ( J  e.  V  ->  J  =  (TopSet `  W ) )
 
Theoremodrngle 13312 The order of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  (  .<_  e.  V  ->  .<_  =  ( le `  W ) )
 
Theoremodrngds 13313 The metric of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  W  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .x.  >. }  u.  {
 <. (TopSet `  ndx ) ,  J >. ,  <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  ( D  e.  V  ->  D  =  ( dist `  W ) )
 
Theoremressds 13314  dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  H  =  ( Gs  A )   &    |-  D  =  (
 dist `  G )   =>    |-  ( A  e.  V  ->  D  =  (
 dist `  H ) )
 
Theoremhomndx 13315 Index value of the df-hom 13228 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  (  Hom  `  ndx )  = ; 1 4
 
Theoremhomid 13316 Utility theorem: index-independent form of df-hom 13228. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- 
 Hom  = Slot  (  Hom  `  ndx )
 
Theoremccondx 13317 Index value of the df-cco 13229 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  (comp `  ndx )  = ; 1
 5
 
Theoremccoid 13318 Utility theorem: index-independent form of df-cco 13229. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- comp  = Slot  (comp `  ndx )
 
Theoremresshom 13319  Hom is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  D  =  ( Cs  A )   &    |-  H  =  ( 
 Hom  `  C )   =>    |-  ( A  e.  V  ->  H  =  (  Hom  `  D ) )
 
Theoremressco 13320 comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  D  =  ( Cs  A )   &    |-  .x.  =  (comp `  C )   =>    |-  ( A  e.  V  ->  .x.  =  (comp `  D ) )
 
7.1.3  Definition of the structure product
 
Syntaxcrest 13321 Extend class notation with the function returning a subspace topology.
 classt
 
Syntaxctopn 13322 Extend class notation with the topology extractor function.
 class  TopOpen
 
Definitiondf-rest 13323* 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 13324 Define the topology extractor function. This differs from df-tset 13223 when a structure has been restricted using df-ress 13151; 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 13325 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
 |-t  Fn  ( _V  X.  _V )
 
Theoremtopnfn 13326 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  TopOpen 
 Fn  _V
 
Theoremrestval 13327* 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 13328* 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 13329 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 13330 Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( (/)t  A )  =  (/)
 
Theoremrestid2 13331 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 13332 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 13333 The finite intersections operator commutes with restriction. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( fi `  ( Jt  A ) )  =  ( ( fi `  J )t  A )
 
Theoremrestid 13334 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 13335 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (TopSet `  W )   =>    |-  ( Jt  B )  =  (
 TopOpen `  W )
 
Theoremtopnid 13336 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 13337 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 13338 Extend class notation with a function that converts a basis to its corresponding topology.
 class  topGen
 
Syntaxcpt 13339 Extend class notation with a function whose value is a product topology.
 class  Xt_
 
Definitiondf-topgen 13340* 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 16690). See tgval3 16697 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 13341* 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 13342 The function constructing structure products.
 class  X_s
 
Syntaxcpws 13343 The function constructing structure powers.
 class  ^s
 
Definitiondf-prds 13344* 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 13345 The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |- 
 Rel  dom  X_s
 
Definitiondf-pws 13346* 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 13347* Lemma for structure products. (Contributed by Mario Carneiro, 3-Jan-2015.)
 |-  B  =  X_ x  e.  dom  R ( Base `  ( R `  x ) )   =>    |-  B  e.  _V
 
Theoremimasvalstr 13348 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 13349 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 13350 Lemma for prdsbas 13353 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 13351* 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 13352 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 13353* 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 13354* 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 13355* 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 13356* 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 13357* 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 13358 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 13359* 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 13360 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 13361 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 13362* 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 13363* 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 13364* 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 13365* 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 13366 Points in the structure product are functions; use this with dffn5 5530 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 13367 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 13368* 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 13369 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 13370* 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 13371 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 13372* 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 13373* 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 13374* 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 13375 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 13376* 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 13377* 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 13378* 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 13379* 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 13380* 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 13381 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 13382 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 13383 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 13384 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 13385 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 13386 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 13387 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 13388* 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 13389 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 13390 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 13391 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 13392 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 13393* 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 13394 Extend class notation with the order topology.
 class ordTop
 
Syntaxcxrs 13395 Extend class notation with the extended real number structure.
 class  RR*
 s
 
Syntaxc0g 13396 Extend class notation with group identity element.
 class  0g
 
Syntaxcgsu 13397 Extend class notation to include finitely supported group sums.
 class  gsumg
 
Definitiondf-ordt 13398* 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 13399* The extended real number structure. Unlike df-cnfld 16374, 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 16374. 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 13400* 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 ) ) ) )
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