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Theorem prdsdsval3 15376
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 2423 . . 3  |-  ( dist `  R )  =  (
dist `  R )
9 prdsdsval3.d . . 3  |-  D  =  ( dist `  Y
)
101, 2, 3, 4, 5, 6, 7, 8, 9prdsdsval2 15375 . 2  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
11 eqidd 2424 . . . . . 6  |-  ( ph  ->  I  =  I )
12 prdsdsval3.k . . . . . . . 8  |-  K  =  ( Base `  R
)
131, 2, 3, 4, 5, 12, 6prdsbascl 15374 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  K )
141, 2, 3, 4, 5, 12, 7prdsbascl 15374 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( G `  x
)  e.  K )
15 prdsdsval3.e . . . . . . . . . . 11  |-  E  =  ( ( dist `  R
)  |`  ( K  X.  K ) )
1615oveqi 6316 . . . . . . . . . 10  |-  ( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( (
dist `  R )  |`  ( K  X.  K
) ) ( G `
 x ) )
17 ovres 6448 . . . . . . . . . 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 2476 . . . . . . . . 9  |-  ( ( ( F `  x
)  e.  K  /\  ( G `  x )  e.  K )  -> 
( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )
1918ex 436 . . . . . . . 8  |-  ( ( F `  x )  e.  K  ->  (
( G `  x
)  e.  K  -> 
( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) ) )
2019ral2imi 2814 . . . . . . 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 63 . . . . . 6  |-  ( ph  ->  A. x  e.  I 
( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )
22 mpteq12 4501 . . . . . 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 666 . . . . 5  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) ( dist `  R
) ( G `  x ) ) ) )
2423rneqd 5079 . . . 4  |-  ( ph  ->  ran  ( x  e.  I  |->  ( ( F `
 x ) E ( G `  x
) ) )  =  ran  ( x  e.  I  |->  ( ( F `
 x ) (
dist `  R )
( G `  x
) ) ) )
2524uneq1d 3620 . . 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 7964 . 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 2467 1  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1869   A.wral 2776    u. cun 3435   {csn 3997    |-> cmpt 4480    X. cxp 4849   ran crn 4852    |` cres 4853   ` cfv 5599  (class class class)co 6303   supcsup 7958   0cc0 9541   RR*cxr 9676    < clt 9677   Basecbs 15114   distcds 15192   X_scprds 15337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-map 7480  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-sup 7960  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-fz 11787  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-mulr 15197  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-hom 15207  df-cco 15208  df-prds 15339
This theorem is referenced by:  prdsxmetlem  21375  prdsmet  21377  prdsbl  21498  prdsbnd  32083  rrnequiv  32125
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