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Theorem prdsdsval3 14546
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 2454 . . 3  |-  ( dist `  R )  =  (
dist `  R )
9 prdsdsval3.d . . 3  |-  D  =  ( dist `  Y
)
101, 2, 3, 4, 5, 6, 7, 8, 9prdsdsval2 14545 . 2  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( x  e.  I  |->  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
11 eqidd 2455 . . . . . 6  |-  ( ph  ->  I  =  I )
12 prdsdsval3.k . . . . . . . 8  |-  K  =  ( Base `  R
)
131, 2, 3, 4, 5, 12, 6prdsbascl 14544 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( F `  x
)  e.  K )
141, 2, 3, 4, 5, 12, 7prdsbascl 14544 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( G `  x
)  e.  K )
15 prdsdsval3.e . . . . . . . . . . 11  |-  E  =  ( ( dist `  R
)  |`  ( K  X.  K ) )
1615oveqi 6216 . . . . . . . . . 10  |-  ( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( (
dist `  R )  |`  ( K  X.  K
) ) ( G `
 x ) )
17 ovres 6343 . . . . . . . . . 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 2507 . . . . . . . . 9  |-  ( ( ( F `  x
)  e.  K  /\  ( G `  x )  e.  K )  -> 
( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )
1918ex 434 . . . . . . . 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 60 . . . . . 6  |-  ( ph  ->  A. x  e.  I 
( ( F `  x ) E ( G `  x ) )  =  ( ( F `  x ) ( dist `  R
) ( G `  x ) ) )
22 mpteq12 4482 . . . . . 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 661 . . . . 5  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) ( dist `  R
) ( G `  x ) ) ) )
2423rneqd 5178 . . . 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 7811 . 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 2498 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 369    = wceq 1370    e. wcel 1758   A.wral 2799    u. cun 3437   {csn 3988    |-> cmpt 4461    X. cxp 4949   ran crn 4952    |` cres 4953   ` cfv 5529  (class class class)co 6203   supcsup 7805   0cc0 9397   RR*cxr 9532    < clt 9533   Basecbs 14296   distcds 14370   X_scprds 14507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-fz 11559  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-plusg 14374  df-mulr 14375  df-sca 14377  df-vsca 14378  df-ip 14379  df-tset 14380  df-ple 14381  df-ds 14383  df-hom 14385  df-cco 14386  df-prds 14509
This theorem is referenced by:  prdsxmetlem  20085  prdsmet  20087  prdsbl  20208  prdsbnd  28863  rrnequiv  28905
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