MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prdsdsval Structured version   Unicode version

Theorem prdsdsval 14428
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 5956 . . . 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 5522 . . . 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 14414 . 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 5702 . . . . . . . 8  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
13 fveq1 5702 . . . . . . . 8  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
1412, 13oveqan12d 6122 . . . . . . 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 4391 . . . . 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 5079 . . . 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 3521 . . 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 7708 . 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 11130 . . . 4  |-  <  Or  RR*
2322supex 7725 . . 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 6230 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 1369    e. wcel 1756   _Vcvv 2984    u. cun 3338   {csn 3889    e. cmpt 4362   dom cdm 4852   ran crn 4853    Fn wfn 5425   ` cfv 5430  (class class class)co 6103   supcsup 7702   0cc0 9294   RR*cxr 9429    < clt 9430   Basecbs 14186   distcds 14259   X_scprds 14396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-fz 11450  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-plusg 14263  df-mulr 14264  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-hom 14274  df-cco 14275  df-prds 14398
This theorem is referenced by:  prdsdsval2  14434  xpsdsval  19968
  Copyright terms: Public domain W3C validator