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Theorem reldmprds 13349
Description: The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
Assertion
Ref Expression
reldmprds  |-  Rel  dom  X_s

Proof of Theorem reldmprds
Dummy variables  a 
c  d  e  f  g  h  s  r  x  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-prds 13348 . 2  |-  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 ) ) ) ) )
>. } ) ) )
21reldmmpt2 5955 1  |-  Rel  dom  X_s
Colors of variables: wff set class
Syntax hints:    /\ wa 358   A.wral 2543   _Vcvv 2788   [_csb 3081    u. cun 3150    C_ wss 3152   {csn 3640   {cpr 3641   {ctp 3642   <.cop 3643   class class class wbr 4023   {copab 4076    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ran crn 4690    o. ccom 4693   Rel wrel 4694   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   X_cixp 6817   supcsup 7193   0cc0 8737   RR*cxr 8866    < clt 8867   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212  TopSetcts 13214   lecple 13215   distcds 13217    Hom chom 13219  compcco 13220   TopOpenctopn 13326   Xt_cpt 13343   X_scprds 13346
This theorem is referenced by:  dsmmval  27200  dsmmval2  27202  dsmmbas2  27203  dsmmfi  27204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-dm 4699  df-oprab 5862  df-mpt2 5863  df-prds 13348
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