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Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempleid 14801 Utility theorem: self-referencing, index-independent form of df-ple 14722. (Contributed by NM, 9-Nov-2012.)
 |- 
 le  = Slot  ( le ` 
 ndx )
 
Theoremotpsstr 14802 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 14803 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 14804 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 14805 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 14806  le is unaffected by restriction. (Contributed by Mario Carneiro, 3-Nov-2015.)
 |-  W  =  ( Ks  A )   &    |-  .<_  =  ( le `  K )   =>    |-  ( A  e.  V  -> 
 .<_  =  ( le `  W ) )
 
Theoremocndx 14807 Index value of the df-ocomp 14723 slot. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  ( oc `  ndx )  = ; 1 1
 
Theoremocid 14808 Utility theorem: index-independent form of df-ocomp 14723. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |- 
 oc  = Slot  ( oc ` 
 ndx )
 
Theoremdsndx 14809 Index value of the df-ds 14724 slot. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( dist `  ndx )  = ; 1
 2
 
Theoremdsid 14810 Utility theorem: index-independent form of df-ds 14724. (Contributed by Mario Carneiro, 23-Dec-2013.)
 |- 
 dist  = Slot  ( dist `  ndx )
 
Theoremunifndx 14811 Index value of the df-unif 14725 slot. (Contributed by Thierry Arnoux, 17-Dec-2017.)
 |-  ( UnifSet `  ndx )  = ; 1
 3
 
Theoremunifid 14812 Utility theorem: index-independent form of df-unif 14725. (Contributed by Thierry Arnoux, 17-Dec-2017.)
 |- 
 UnifSet  = Slot  ( UnifSet `  ndx )
 
Theoremodrngstr 14813 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 14814 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 14815 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 14816 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 14817 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 14818 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 14819 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 14820  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 14821 Index value of the df-hom 14726 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  ( Hom  `  ndx )  = ; 1 4
 
Theoremhomid 14822 Utility theorem: index-independent form of df-hom 14726. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- 
 Hom  = Slot  ( Hom  `  ndx )
 
Theoremccondx 14823 Index value of the df-cco 14727 slot. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |-  (comp `  ndx )  = ; 1
 5
 
Theoremccoid 14824 Utility theorem: index-independent form of df-cco 14727. (Contributed by Mario Carneiro, 7-Jan-2017.)
 |- comp  = Slot  (comp `  ndx )
 
Theoremresshom 14825  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 14826 comp is unaffected by restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  D  =  ( Cs  A )   &    |-  .x.  =  (comp `  C )   =>    |-  ( A  e.  V  ->  .x.  =  (comp `  D ) )
 
Theoremslotsbhcdif 14827 The slots  Base,  Hom and comp are different. (Contributed by AV, 5-Mar-2020.)
 |-  ( ( Base `  ndx )  =/=  ( Hom  `  ndx )  /\  ( Base `  ndx )  =/=  (comp `  ndx )  /\  ( Hom  `  ndx )  =/=  (comp `  ndx ) )
 
7.1.3  Definition of the structure product
 
Syntaxcrest 14828 Extend class notation with the function returning a subspace topology.
 classt
 
Syntaxctopn 14829 Extend class notation with the topology extractor function.
 class  TopOpen
 
Definitiondf-rest 14830* 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 14831 Define the topology extractor function. This differs from df-tset 14721 when a structure has been restricted using df-ress 14641; 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 14832 The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
 |-t  Fn  ( _V  X.  _V )
 
Theoremtopnfn 14833 The topology extractor function is a function on the universe. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  TopOpen 
 Fn  _V
 
Theoremrestval 14834* 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 14835* 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 14836 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 14837 Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( (/)t  A )  =  (/)
 
Theoremrestid2 14838 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 14839 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 14840 The finite intersections operator commutes with restriction. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( fi `  ( Jt  A ) )  =  ( ( fi `  J )t  A )
 
Theoremrestid 14841 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 14842 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  B  =  ( Base `  W )   &    |-  J  =  (TopSet `  W )   =>    |-  ( Jt  B )  =  (
 TopOpen `  W )
 
Theoremtopnid 14843 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 14844 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 14845 Extend class notation with a function that converts a basis to its corresponding topology.
 class  topGen
 
Syntaxcpt 14846 Extend class notation with a function whose value is a product topology.
 class  Xt_
 
Syntaxc0g 14847 Extend class notation with group identity element.
 class  0g
 
Syntaxcgsu 14848 Extend class notation to include finitely supported group sums.
 class  gsumg
 
Definitiondf-0g 14849* Define group identity element. Remark: this definition is required here because the symbol  0g is already used in df-gsum 14850. The related theorems are provided later, see grpidval 16004. (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 14850* 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 different 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 20710. Remark: this definition is required here because the symbol  gsumg is already used in df-prds 14855 and df-imas 14915. The related theorems are provided later, see gsumvalx 16014. (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 ) ) ) ) ) ) )
 
Definitiondf-topgen 14851* 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 19542). See tgval3 19549 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 14852* 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 14853 The function constructing structure products.
 class  X_s
 
Syntaxcpws 14854 The function constructing structure powers.
 class  ^s
 
Definitiondf-prds 14855* 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 is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
 |-  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 ) ) ) ) >. ,  <. ( .i
 `  ndx ) ,  (
 f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `
  x ) ( .i `  ( 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 14856 The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
 |- 
 Rel  dom  X_s
 
Definitiondf-pws 14857* 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 14858* Lemma for structure products. (Contributed by Mario Carneiro, 3-Jan-2015.)
 |-  B  =  X_ x  e.  dom  R ( Base `  ( R `  x ) )   =>    |-  B  e.  _V
 
Theoremimasvalstr 14859 Structure product value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
 <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )  u.  { <. (TopSet `  ndx ) ,  O >. ,  <. ( le ` 
 ndx ) ,  L >. ,  <. ( dist `  ndx ) ,  D >. } )   =>    |-  U Struct  <. 1 , ; 1 2 >.
 
Theoremprdsvalstr 14860 Structure product value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  ( ( { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )  u.  ( { <. (TopSet `  ndx ) ,  O >. , 
 <. ( le `  ndx ) ,  L >. , 
 <. ( dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .xb  >. } ) ) Struct  <. 1 , ; 1 5 >.
 
Theoremprdsvallem 14861 Lemma for prdsbas 14864 and similar theorems. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  ( ph  ->  U  =  ( ( { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  {
 <. (Scalar `  ndx ) ,  S >. ,  <. ( .s
 `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )  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.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )  u.  ( { <. (TopSet `  ndx ) ,  O >. , 
 <. ( le `  ndx ) ,  L >. , 
 <. ( dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .xb  >. } ) )   =>    |-  ( ph  ->  A  =  T )
 
Theoremprdsval 14862* Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  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  ->  .,  =  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i `  ( 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.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )  u.  ( { <. (TopSet `  ndx ) ,  O >. , 
 <. ( le `  ndx ) ,  .<_  >. ,  <. (
 dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .xb  >. } ) ) )
 
Theoremprdssca 14863 Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   =>    |-  ( ph  ->  S  =  (Scalar `  P )
 )
 
Theoremprdsbas 14864* Base set of a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  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 14865* Addition in a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  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 14866* Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  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 14867* Scalar multiplication in a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  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 ) ) ) ) )
 
Theoremprdsip 14868* Inner product in a structure product. (Contributed by Thierry Arnoux, 16-Jun-2019.)
 |-  P  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R  e.  W )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  dom 
 R  =  I )   &    |-  .,  =  ( .i `  P )   =>    |-  ( ph  ->  .,  =  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
 `  ( R `  x ) ) ( g `  x ) ) ) ) ) )
 
Theoremprdsle 14869* Structure product weak ordering. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  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 14870 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 14871* Structure product distance function. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  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 14872 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 14873 Structure product topology. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  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 14874* Structure product hom-sets. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  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 14875* Structure product composition operation. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
 |-  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 14876* 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 14877* 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 14878 Points in the structure product are functions; use this with dffn5 5819 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 14879 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 14880* 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 14881 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 14882* 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 14883 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 14884* 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 14885* 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 14886* 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 14887 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 14888* 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 14889* 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 14890* 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 14891* 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 14892* 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 14893 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 14894 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 14895 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 14896 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 14897 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  oF  .+  G ) )
 
Theorempwsmulrval 14898 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  oF  .x.  G ) )
 
Theorempwsle 14899 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 )  ->  .<_  =  (  oR O  i^i  ( B  X.  B ) ) )
 
Theorempwsleval 14900* 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 ) ) )
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