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Theorem prdsdsval 14894
Description: Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
prdsbasmpt.y  |-  Y  =  ( S X_s R )
prdsbasmpt.b  |-  B  =  ( Base `  Y
)
prdsbasmpt.s  |-  ( ph  ->  S  e.  V )
prdsbasmpt.i  |-  ( ph  ->  I  e.  W )
prdsbasmpt.r  |-  ( ph  ->  R  Fn  I )
prdsplusgval.f  |-  ( ph  ->  F  e.  B )
prdsplusgval.g  |-  ( ph  ->  G  e.  B )
prdsdsval.d  |-  D  =  ( dist `  Y
)
Assertion
Ref Expression
prdsdsval  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  ( R `  x )
) ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
Distinct variable groups:    x, B    x, F    x, G    ph, x    x, I    x, V    x, R    x, S    x, W    x, Y
Allowed substitution hint:    D( x)

Proof of Theorem prdsdsval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbasmpt.y . . 3  |-  Y  =  ( S X_s R )
2 prdsbasmpt.s . . 3  |-  ( ph  ->  S  e.  V )
3 prdsbasmpt.r . . . 4  |-  ( ph  ->  R  Fn  I )
4 prdsbasmpt.i . . . 4  |-  ( ph  ->  I  e.  W )
5 fnex 6140 . . . 4  |-  ( ( R  Fn  I  /\  I  e.  W )  ->  R  e.  _V )
63, 4, 5syl2anc 661 . . 3  |-  ( ph  ->  R  e.  _V )
7 prdsbasmpt.b . . 3  |-  B  =  ( Base `  Y
)
8 fndm 5686 . . . 4  |-  ( R  Fn  I  ->  dom  R  =  I )
93, 8syl 16 . . 3  |-  ( ph  ->  dom  R  =  I )
10 prdsdsval.d . . 3  |-  D  =  ( dist `  Y
)
111, 2, 6, 7, 9, 10prdsds 14880 . 2  |-  ( ph  ->  D  =  ( f  e.  B ,  g  e.  B  |->  sup (
( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) )
12 fveq1 5871 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
13 fveq1 5871 . . . . . . . 8  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
1412, 13oveqan12d 6315 . . . . . . 7  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) )  =  ( ( F `
 x ) (
dist `  ( R `  x ) ) ( G `  x ) ) )
1514adantl 466 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) )  =  ( ( F `
 x ) (
dist `  ( R `  x ) ) ( G `  x ) ) )
1615mpteq2dv 4544 . . . . 5  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )  =  ( x  e.  I  |->  ( ( F `  x ) ( dist `  ( R `  x )
) ( G `  x ) ) ) )
1716rneqd 5240 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )  =  ran  (
x  e.  I  |->  ( ( F `  x
) ( dist `  ( R `  x )
) ( G `  x ) ) ) )
1817uneq1d 3653 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } )  =  ( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  ( R `  x
) ) ( G `
 x ) ) )  u.  { 0 } ) )
1918supeq1d 7923 . 2  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  sup ( ( ran  (
x  e.  I  |->  ( ( f `  x
) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  )  =  sup ( ( ran  ( x  e.  I  |->  ( ( F `
 x ) (
dist `  ( R `  x ) ) ( G `  x ) ) )  u.  {
0 } ) , 
RR* ,  <  ) )
20 prdsplusgval.f . 2  |-  ( ph  ->  F  e.  B )
21 prdsplusgval.g . 2  |-  ( ph  ->  G  e.  B )
22 xrltso 11372 . . . 4  |-  <  Or  RR*
2322supex 7940 . . 3  |-  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  ( R `  x )
) ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  )  e.  _V
2423a1i 11 . 2  |-  ( ph  ->  sup ( ( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  ( R `  x
) ) ( G `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  )  e.  _V )
2511, 19, 20, 21, 24ovmpt2d 6429 1  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  ( R `  x )
) ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469   {csn 4032    |-> cmpt 4515   dom cdm 5008   ran crn 5009    Fn wfn 5589   ` cfv 5594  (class class class)co 6296   supcsup 7918   0cc0 9509   RR*cxr 9644    < clt 9645   Basecbs 14643   distcds 14720   X_scprds 14862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-hom 14735  df-cco 14736  df-prds 14864
This theorem is referenced by:  prdsdsval2  14900  xpsdsval  21009
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