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Theorem prdsdsval3 13699
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 2435 . . 3  |-  ( dist `  R )  =  (
dist `  R )
9 prdsdsval3.d . . 3  |-  D  =  ( dist `  Y
)
101, 2, 3, 4, 5, 6, 7, 8, 9prdsdsval2 13698 . 2  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
11 eqidd 2436 . . . . . 6  |-  ( ph  ->  I  =  I )
12 prdsdsval3.k . . . . . . . 8  |-  K  =  ( Base `  R
)
131, 2, 3, 4, 5, 12, 6prdsbascl 13697 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  K )
141, 2, 3, 4, 5, 12, 7prdsbascl 13697 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( G `  x
)  e.  K )
15 prdsdsval3.e . . . . . . . . . . 11  |-  E  =  ( ( dist `  R
)  |`  ( K  X.  K ) )
1615oveqi 6086 . . . . . . . . . 10  |-  ( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( (
dist `  R )  |`  ( K  X.  K
) ) ( G `
 x ) )
17 ovres 6205 . . . . . . . . . 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 2479 . . . . . . . . 9  |-  ( ( ( F `  x
)  e.  K  /\  ( G `  x )  e.  K )  -> 
( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )
1918ex 424 . . . . . . . 8  |-  ( ( F `  x )  e.  K  ->  (
( G `  x
)  e.  K  -> 
( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) ) )
2019ral2imi 2774 . . . . . . 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 58 . . . . . 6  |-  ( ph  ->  A. x  e.  I 
( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )
22 mpteq12 4280 . . . . . 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 643 . . . . 5  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) ( dist `  R
) ( G `  x ) ) ) )
2423rneqd 5089 . . . 4  |-  ( ph  ->  ran  ( x  e.  I  |->  ( ( F `
 x ) E ( G `  x
) ) )  =  ran  ( x  e.  I  |->  ( ( F `
 x ) (
dist `  R )
( G `  x
) ) ) )
2524uneq1d 3492 . . 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 7443 . 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 2470 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 359    = wceq 1652    e. wcel 1725   A.wral 2697    u. cun 3310   {csn 3806    e. cmpt 4258    X. cxp 4868   ran crn 4871    |` cres 4872   ` cfv 5446  (class class class)co 6073   supcsup 7437   0cc0 8982   RR*cxr 9111    < clt 9112   Basecbs 13461   distcds 13530   X_scprds 13661
This theorem is referenced by:  prdsxmetlem  18390  prdsmet  18392  prdsbl  18513  prdsbnd  26493  rrnequiv  26535
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-hom 13545  df-cco 13546  df-prds 13663
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