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Theorem reldmprds 13672
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 13671 . 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 6181 1  |-  Rel  dom  X_s
Colors of variables: wff set class
Syntax hints:    /\ wa 359   A.wral 2705   _Vcvv 2956   [_csb 3251    u. cun 3318    C_ wss 3320   {csn 3814   {cpr 3815   {ctp 3816   <.cop 3817   class class class wbr 4212   {copab 4265    e. cmpt 4266    X. cxp 4876   dom cdm 4878   ran crn 4879    o. ccom 4882   Rel wrel 4883   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   X_cixp 7063   supcsup 7445   0cc0 8990   RR*cxr 9119    < clt 9120   ndxcnx 13466   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533  TopSetcts 13535   lecple 13536   distcds 13538    Hom chom 13540  compcco 13541   TopOpenctopn 13649   Xt_cpt 13666   X_scprds 13669
This theorem is referenced by:  dsmmval  27177  dsmmval2  27179  dsmmbas2  27180  dsmmfi  27181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-dm 4888  df-oprab 6085  df-mpt2 6086  df-prds 13671
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