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Theorem reldmprds 14955
Description: The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.)
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 14954 . 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 ) ) ) ) >. ,  <. ( .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 ) ) ) ) )
>. } ) ) )
21reldmmpt2 6350 1  |-  Rel  dom  X_s
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367   A.wral 2753   _Vcvv 3058   [_csb 3372    u. cun 3411    C_ wss 3413   {csn 3971   {cpr 3973   {ctp 3975   <.cop 3977   class class class wbr 4394   {copab 4451    |-> cmpt 4452    X. cxp 4940   dom cdm 4942   ran crn 4943    o. ccom 4946   Rel wrel 4947   ` cfv 5525  (class class class)co 6234    |-> cmpt2 6236   1stc1st 6736   2ndc2nd 6737   X_cixp 7427   supcsup 7854   0cc0 9442   RR*cxr 9577    < clt 9578   ndxcnx 14730   Basecbs 14733   +g cplusg 14801   .rcmulr 14802  Scalarcsca 14804   .scvsca 14805   .icip 14806  TopSetcts 14807   lecple 14808   distcds 14810   Hom chom 14812  compcco 14813   TopOpenctopn 14928   Xt_cpt 14945    gsumg cgsu 14947   X_scprds 14952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4948  df-rel 4949  df-dm 4952  df-oprab 6238  df-mpt2 6239  df-prds 14954
This theorem is referenced by:  dsmmval  18955  dsmmval2  18957  dsmmbas2  18958  dsmmfi  18959
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