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Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempleid 13301 Utility theorem: self-referencing, index-independent form of df-ple 13228. (Contributed by NM, 9-Nov-2012.)
 |- 
 le  = Slot  ( le ` 
 ndx )
 
Theoremotpsstr 13302 Functionality of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  B >. ,  <. (TopSet `  ndx ) ,  J >. , 
 <. ( le `  ndx ) ,  .<_  >. }   =>    |-  K Struct  <. 1 ,  10 >.
 
Theoremotpsbas 13303 The base set of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  B >. ,  <. (TopSet `  ndx ) ,  J >. , 
 <. ( le `  ndx ) ,  .<_  >. }   =>    |-  ( B  e.  V  ->  B  =  (
 Base `  K ) )
 
Theoremotpstset 13304 The open sets of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.)
 |-  K  =  { <. (
 Base `  ndx ) ,  B >. ,  <. (TopSet `  ndx ) ,  J >. , 
 <. ( le `  ndx ) ,  .<_  >. }   =>    |-  ( J  e.  V  ->  J  =  (TopSet `  K ) )
 
Theoremotpsle 13305 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 13306  le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
 |-  W  =  ( Ks  A )   &    |-  .<_  =  ( le `  K )   =>    |-  ( A  e.  V  -> 
 .<_  =  ( le `  W ) )
 
Theoremocndx 13307 Index value of the df-ocomp 13229 slot. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  ( oc `  ndx )  = ; 1 1
 
Theoremocid 13308 Utility theorem: index-independent form of df-ocomp 13229. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |- 
 oc  = Slot  ( oc ` 
 ndx )
 
Theoremdsndx 13309 Index value of the df-ds 13230 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( dist `  ndx )  = ; 1
 2
 
Theoremdsid 13310 Utility theorem: index-independent form of df-ds 13230. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |- 
 dist  = Slot  ( dist `  ndx )
 
Theoremodrngstr 13311 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 13312 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 13313 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 13314 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 13315 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 13316 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 13317 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 13318  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 13319 Index value of the df-hom 13232 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  (  Hom  `  ndx )  = ; 1 4
 
Theoremhomid 13320 Utility theorem: index-independent form of df-hom 13232. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- 
 Hom  = Slot  (  Hom  `  ndx )
 
Theoremccondx 13321 Index value of the df-cco 13233 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  (comp `  ndx )  = ; 1
 5
 
Theoremccoid 13322 Utility theorem: index-independent form of df-cco 13233. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- comp  = Slot  (comp `  ndx )
 
Theoremresshom 13323  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 13324 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 13325 Extend class notation with the function returning a subspace topology.
 classt
 
Syntaxctopn 13326 Extend class notation with the topology extractor function.
 class  TopOpen
 
Definitiondf-rest 13327* 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 13328 Define the topology extractor function. This differs from df-tset 13227 when a structure has been restricted using df-ress 13155; 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 13329 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
 |-t  Fn  ( _V  X.  _V )
 
Theoremtopnfn 13330 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  TopOpen 
 Fn  _V
 
Theoremrestval 13331* 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 13332* 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 13333 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 13334 Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( (/)t  A )  =  (/)
 
Theoremrestid2 13335 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 13336 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 13337 The finite intersections operator commutes with restriction. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( fi `  ( Jt  A ) )  =  ( ( fi `  J )t  A )
 
Theoremrestid 13338 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 13339 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (TopSet `  W )   =>    |-  ( Jt  B )  =  (
 TopOpen `  W )
 
Theoremtopnid 13340 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 13341 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 13342 Extend class notation with a function that converts a basis to its corresponding topology.
 class  topGen
 
Syntaxcpt 13343 Extend class notation with a function whose value is a product topology.
 class  Xt_
 
Definitiondf-topgen 13344* 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 16694). See tgval3 16701 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 13345* 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 13346 The function constructing structure products.
 class  X_s
 
Syntaxcpws 13347 The function constructing structure powers.
 class  ^s
 
Definitiondf-prds 13348* 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 13349 The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |- 
 Rel  dom  X_s
 
Definitiondf-pws 13350* 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 13351* Lemma for structure products. (Contributed by Mario Carneiro, 3-Jan-2015.)
 |-  B  =  X_ x  e.  dom  R ( Base `  ( R `  x ) )   =>    |-  B  e.  _V
 
Theoremimasvalstr 13352 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 13353 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 13354 Lemma for prdsbas 13357 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 13355* 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 13356 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 13357* 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 13358* 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 13359* 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 13360* 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 13361* 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 13362 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 13363* 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 13364 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 13365 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 13366* 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 13367* 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 13368* 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 13369* 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 13370 Points in the structure product are functions; use this with dffn5 5568 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 13371 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 13372* 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 13373 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 13374* 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 13375 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 13376* 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 13377* 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 13378* 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 13379 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 13380* 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 13381* 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 13382* 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 13383* 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 13384* 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 13385 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 13386 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 13387 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 13388 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 13389 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 13390 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 13391 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 13392* 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 13393 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 13394 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 13395 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 13396 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 13397* 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 13398 Extend class notation with the order topology.
 class ordTop
 
Syntaxcxrs 13399 Extend class notation with the extended real number structure.
 class  RR*
 s
 
Syntaxc0g 13400 Extend class notation with group identity element.
 class  0g
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