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Theorem prdsdsval3 13384
Description: Value of the metric in a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
Hypotheses
Ref Expression
prdsbasmpt2.y  |-  Y  =  ( S X_s ( x  e.  I  |->  R ) )
prdsbasmpt2.b  |-  B  =  ( Base `  Y
)
prdsbasmpt2.s  |-  ( ph  ->  S  e.  V )
prdsbasmpt2.i  |-  ( ph  ->  I  e.  W )
prdsbasmpt2.r  |-  ( ph  ->  A. x  e.  I  R  e.  X )
prdsdsval2.f  |-  ( ph  ->  F  e.  B )
prdsdsval2.g  |-  ( ph  ->  G  e.  B )
prdsdsval3.k  |-  K  =  ( Base `  R
)
prdsdsval3.e  |-  E  =  ( ( dist `  R
)  |`  ( K  X.  K ) )
prdsdsval3.d  |-  D  =  ( dist `  Y
)
Assertion
Ref Expression
prdsdsval3  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
Distinct variable groups:    x, F    x, G    x, I
Allowed substitution hints:    ph( x)    B( x)    D( x)    R( x)    S( x)    E( x)    K( x)    V( x)    W( x)    X( x)    Y( x)

Proof of Theorem prdsdsval3
StepHypRef Expression
1 prdsbasmpt2.y . . 3  |-  Y  =  ( S X_s ( x  e.  I  |->  R ) )
2 prdsbasmpt2.b . . 3  |-  B  =  ( Base `  Y
)
3 prdsbasmpt2.s . . 3  |-  ( ph  ->  S  e.  V )
4 prdsbasmpt2.i . . 3  |-  ( ph  ->  I  e.  W )
5 prdsbasmpt2.r . . 3  |-  ( ph  ->  A. x  e.  I  R  e.  X )
6 prdsdsval2.f . . 3  |-  ( ph  ->  F  e.  B )
7 prdsdsval2.g . . 3  |-  ( ph  ->  G  e.  B )
8 eqid 2283 . . 3  |-  ( dist `  R )  =  (
dist `  R )
9 prdsdsval3.d . . 3  |-  D  =  ( dist `  Y
)
101, 2, 3, 4, 5, 6, 7, 8, 9prdsdsval2 13383 . 2  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
11 eqidd 2284 . . . . . 6  |-  ( ph  ->  I  =  I )
12 prdsdsval3.k . . . . . . . 8  |-  K  =  ( Base `  R
)
131, 2, 3, 4, 5, 12, 6prdsbascl 13382 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  K )
141, 2, 3, 4, 5, 12, 7prdsbascl 13382 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( G `  x
)  e.  K )
15 prdsdsval3.e . . . . . . . . . . 11  |-  E  =  ( ( dist `  R
)  |`  ( K  X.  K ) )
1615oveqi 5871 . . . . . . . . . 10  |-  ( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( (
dist `  R )  |`  ( K  X.  K
) ) ( G `
 x ) )
17 ovres 5987 . . . . . . . . . 10  |-  ( ( ( F `  x
)  e.  K  /\  ( G `  x )  e.  K )  -> 
( ( F `  x ) ( (
dist `  R )  |`  ( K  X.  K
) ) ( G `
 x ) )  =  ( ( F `
 x ) (
dist `  R )
( G `  x
) ) )
1816, 17syl5eq 2327 . . . . . . . . 9  |-  ( ( ( F `  x
)  e.  K  /\  ( G `  x )  e.  K )  -> 
( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )
1918ex 423 . . . . . . . 8  |-  ( ( F `  x )  e.  K  ->  (
( G `  x
)  e.  K  -> 
( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) ) )
2019ral2imi 2619 . . . . . . 7  |-  ( A. x  e.  I  ( F `  x )  e.  K  ->  ( A. x  e.  I  ( G `  x )  e.  K  ->  A. x  e.  I  ( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R ) ( G `
 x ) ) ) )
2113, 14, 20sylc 56 . . . . . 6  |-  ( ph  ->  A. x  e.  I 
( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )
22 mpteq12 4099 . . . . . 6  |-  ( ( I  =  I  /\  A. x  e.  I  ( ( F `  x
) E ( G `
 x ) )  =  ( ( F `
 x ) (
dist `  R )
( G `  x
) ) )  -> 
( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) ( dist `  R
) ( G `  x ) ) ) )
2311, 21, 22syl2anc 642 . . . . 5  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) ( dist `  R
) ( G `  x ) ) ) )
2423rneqd 4906 . . . 4  |-  ( ph  ->  ran  ( x  e.  I  |->  ( ( F `
 x ) E ( G `  x
) ) )  =  ran  ( x  e.  I  |->  ( ( F `
 x ) (
dist `  R )
( G `  x
) ) ) )
2524uneq1d 3328 . . 3  |-  ( ph  ->  ( ran  ( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) )  u.  { 0 } )  =  ( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  R ) ( G `
 x ) ) )  u.  { 0 } ) )
2625supeq1d 7199 . 2  |-  ( ph  ->  sup ( ( ran  ( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) )  u.  {
0 } ) , 
RR* ,  <  )  =  sup ( ( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  R ) ( G `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
2710, 26eqtr4d 2318 1  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    u. cun 3150   {csn 3640    e. cmpt 4077    X. cxp 4687   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858   supcsup 7193   0cc0 8737   RR*cxr 8866    < clt 8867   Basecbs 13148   distcds 13217   X_scprds 13346
This theorem is referenced by:  prdsxmetlem  17932  prdsmet  17934  prdsbl  18037  prdsbnd  26517  rrnequiv  26559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348
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