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Theorem prdsdsval 13700
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 5961 . . . 4  |-  ( ( R  Fn  I  /\  I  e.  W )  ->  R  e.  _V )
63, 4, 5syl2anc 643 . . 3  |-  ( ph  ->  R  e.  _V )
7 prdsbasmpt.b . . 3  |-  B  =  ( Base `  Y
)
8 fndm 5544 . . . 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 13686 . 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 5727 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
13 fveq1 5727 . . . . . . . 8  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
1412, 13oveqan12d 6100 . . . . . . 7  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) )  =  ( ( F `
 x ) (
dist `  ( R `  x ) ) ( G `  x ) ) )
1514adantl 453 . . . . . 6  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) )  =  ( ( F `
 x ) (
dist `  ( R `  x ) ) ( G `  x ) ) )
1615mpteq2dv 4296 . . . . 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 5097 . . . 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 3500 . . 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 7451 . 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 10734 . . . 4  |-  <  Or  RR*
2322supex 7468 . . 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 6201 1  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  ( R `  x )
) ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    u. cun 3318   {csn 3814    e. cmpt 4266   dom cdm 4878   ran crn 4879    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   supcsup 7445   0cc0 8990   RR*cxr 9119    < clt 9120   Basecbs 13469   distcds 13538   X_scprds 13669
This theorem is referenced by:  prdsdsval2  13706  xpsdsval  18411
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-hom 13553  df-cco 13554  df-prds 13671
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