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Theorem prdsdsval3 13628
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 2381 . . 3  |-  ( dist `  R )  =  (
dist `  R )
9 prdsdsval3.d . . 3  |-  D  =  ( dist `  Y
)
101, 2, 3, 4, 5, 6, 7, 8, 9prdsdsval2 13627 . 2  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
11 eqidd 2382 . . . . . 6  |-  ( ph  ->  I  =  I )
12 prdsdsval3.k . . . . . . . 8  |-  K  =  ( Base `  R
)
131, 2, 3, 4, 5, 12, 6prdsbascl 13626 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  K )
141, 2, 3, 4, 5, 12, 7prdsbascl 13626 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( G `  x
)  e.  K )
15 prdsdsval3.e . . . . . . . . . . 11  |-  E  =  ( ( dist `  R
)  |`  ( K  X.  K ) )
1615oveqi 6027 . . . . . . . . . 10  |-  ( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( (
dist `  R )  |`  ( K  X.  K
) ) ( G `
 x ) )
17 ovres 6146 . . . . . . . . . 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 2425 . . . . . . . . 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 2719 . . . . . . 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 4223 . . . . . 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 5031 . . . 4  |-  ( ph  ->  ran  ( x  e.  I  |->  ( ( F `
 x ) E ( G `  x
) ) )  =  ran  ( x  e.  I  |->  ( ( F `
 x ) (
dist `  R )
( G `  x
) ) ) )
2524uneq1d 3437 . . 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 7380 . 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 2416 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 1649    e. wcel 1717   A.wral 2643    u. cun 3255   {csn 3751    e. cmpt 4201    X. cxp 4810   ran crn 4813    |` cres 4814   ` cfv 5388  (class class class)co 6014   supcsup 7374   0cc0 8917   RR*cxr 9046    < clt 9047   Basecbs 13390   distcds 13459   X_scprds 13590
This theorem is referenced by:  prdsxmetlem  18300  prdsmet  18302  prdsbl  18405  prdsbnd  26187  rrnequiv  26229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-cnex 8973  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-pre-mulgt0 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-int 3987  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-riota 6479  df-recs 6563  df-rdg 6598  df-1o 6654  df-oadd 6658  df-er 6835  df-map 6950  df-ixp 6994  df-en 7040  df-dom 7041  df-sdom 7042  df-fin 7043  df-sup 7375  df-pnf 9049  df-mnf 9050  df-xr 9051  df-ltxr 9052  df-le 9053  df-sub 9219  df-neg 9220  df-nn 9927  df-2 9984  df-3 9985  df-4 9986  df-5 9987  df-6 9988  df-7 9989  df-8 9990  df-9 9991  df-10 9992  df-n0 10148  df-z 10209  df-dec 10309  df-uz 10415  df-fz 10970  df-struct 13392  df-ndx 13393  df-slot 13394  df-base 13395  df-plusg 13463  df-mulr 13464  df-sca 13466  df-vsca 13467  df-tset 13469  df-ple 13470  df-ds 13472  df-hom 13474  df-cco 13475  df-prds 13592
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