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Theorem prdsdsval2 15341
Description: Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-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 )
prdsdsval2.e  |-  E  =  ( dist `  R
)
prdsdsval2.d  |-  D  =  ( dist `  Y
)
Assertion
Ref Expression
prdsdsval2  |-  ( 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)    V( x)    W( x)    X( x)    Y( x)

Proof of Theorem prdsdsval2
Dummy variable  y is distinct from all other variables.
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 . . . 4  |-  ( ph  ->  A. x  e.  I  R  e.  X )
6 eqid 2429 . . . . 5  |-  ( x  e.  I  |->  R )  =  ( x  e.  I  |->  R )
76fnmpt 5722 . . . 4  |-  ( A. x  e.  I  R  e.  X  ->  ( x  e.  I  |->  R )  Fn  I )
85, 7syl 17 . . 3  |-  ( ph  ->  ( x  e.  I  |->  R )  Fn  I
)
9 prdsdsval2.f . . 3  |-  ( ph  ->  F  e.  B )
10 prdsdsval2.g . . 3  |-  ( ph  ->  G  e.  B )
11 prdsdsval2.d . . 3  |-  D  =  ( dist `  Y
)
121, 2, 3, 4, 8, 9, 10, 11prdsdsval 15335 . 2  |-  ( ph  ->  ( F D G )  =  sup (
( ran  ( y  e.  I  |->  ( ( F `  y ) ( dist `  (
( x  e.  I  |->  R ) `  y
) ) ( G `
 y ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
13 nfcv 2591 . . . . . . . 8  |-  F/_ x
( F `  y
)
14 nfcv 2591 . . . . . . . . 9  |-  F/_ x dist
15 nffvmpt1 5889 . . . . . . . . 9  |-  F/_ x
( ( x  e.  I  |->  R ) `  y )
1614, 15nffv 5888 . . . . . . . 8  |-  F/_ x
( dist `  ( (
x  e.  I  |->  R ) `  y ) )
17 nfcv 2591 . . . . . . . 8  |-  F/_ x
( G `  y
)
1813, 16, 17nfov 6331 . . . . . . 7  |-  F/_ x
( ( F `  y ) ( dist `  ( ( x  e.  I  |->  R ) `  y ) ) ( G `  y ) )
19 nfcv 2591 . . . . . . 7  |-  F/_ y
( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) )
20 fveq2 5881 . . . . . . . . 9  |-  ( y  =  x  ->  (
( x  e.  I  |->  R ) `  y
)  =  ( ( x  e.  I  |->  R ) `  x ) )
2120fveq2d 5885 . . . . . . . 8  |-  ( y  =  x  ->  ( dist `  ( ( x  e.  I  |->  R ) `
 y ) )  =  ( dist `  (
( x  e.  I  |->  R ) `  x
) ) )
22 fveq2 5881 . . . . . . . 8  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
23 fveq2 5881 . . . . . . . 8  |-  ( y  =  x  ->  ( G `  y )  =  ( G `  x ) )
2421, 22, 23oveq123d 6326 . . . . . . 7  |-  ( y  =  x  ->  (
( F `  y
) ( dist `  (
( x  e.  I  |->  R ) `  y
) ) ( G `
 y ) )  =  ( ( F `
 x ) (
dist `  ( (
x  e.  I  |->  R ) `  x ) ) ( G `  x ) ) )
2518, 19, 24cbvmpt 4517 . . . . . 6  |-  ( y  e.  I  |->  ( ( F `  y ) ( dist `  (
( x  e.  I  |->  R ) `  y
) ) ( G `
 y ) ) )  =  ( x  e.  I  |->  ( ( F `  x ) ( dist `  (
( x  e.  I  |->  R ) `  x
) ) ( G `
 x ) ) )
26 eqidd 2430 . . . . . . 7  |-  ( ph  ->  I  =  I )
276fvmpt2 5973 . . . . . . . . . . . 12  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( ( x  e.  I  |->  R ) `  x )  =  R )
2827fveq2d 5885 . . . . . . . . . . 11  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( dist `  (
( x  e.  I  |->  R ) `  x
) )  =  (
dist `  R )
)
29 prdsdsval2.e . . . . . . . . . . 11  |-  E  =  ( dist `  R
)
3028, 29syl6eqr 2488 . . . . . . . . . 10  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( dist `  (
( x  e.  I  |->  R ) `  x
) )  =  E )
3130oveqd 6322 . . . . . . . . 9  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) )  =  ( ( F `  x ) E ( G `  x ) ) )
3231ralimiaa 2824 . . . . . . . 8  |-  ( A. x  e.  I  R  e.  X  ->  A. x  e.  I  ( ( F `  x )
( dist `  ( (
x  e.  I  |->  R ) `  x ) ) ( G `  x ) )  =  ( ( F `  x ) E ( G `  x ) ) )
335, 32syl 17 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) )  =  ( ( F `  x ) E ( G `  x ) ) )
34 mpteq12 4505 . . . . . . 7  |-  ( ( I  =  I  /\  A. x  e.  I  ( ( F `  x
) ( dist `  (
( x  e.  I  |->  R ) `  x
) ) ( G `
 x ) )  =  ( ( F `
 x ) E ( G `  x
) ) )  -> 
( x  e.  I  |->  ( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) E ( G `
 x ) ) ) )
3526, 33, 34syl2anc 665 . . . . . 6  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) E ( G `
 x ) ) ) )
3625, 35syl5eq 2482 . . . . 5  |-  ( ph  ->  ( y  e.  I  |->  ( ( F `  y ) ( dist `  ( ( x  e.  I  |->  R ) `  y ) ) ( G `  y ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) E ( G `
 x ) ) ) )
3736rneqd 5082 . . . 4  |-  ( ph  ->  ran  ( y  e.  I  |->  ( ( F `
 y ) (
dist `  ( (
x  e.  I  |->  R ) `  y ) ) ( G `  y ) ) )  =  ran  ( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) ) )
3837uneq1d 3625 . . 3  |-  ( ph  ->  ( ran  ( y  e.  I  |->  ( ( F `  y ) ( dist `  (
( x  e.  I  |->  R ) `  y
) ) ( G `
 y ) ) )  u.  { 0 } )  =  ( ran  ( x  e.  I  |->  ( ( F `
 x ) E ( G `  x
) ) )  u. 
{ 0 } ) )
3938supeq1d 7966 . 2  |-  ( ph  ->  sup ( ( ran  ( y  e.  I  |->  ( ( F `  y ) ( dist `  ( ( x  e.  I  |->  R ) `  y ) ) ( G `  y ) ) )  u.  {
0 } ) , 
RR* ,  <  )  =  sup ( ( ran  ( x  e.  I  |->  ( ( F `  x ) E ( G `  x ) ) )  u.  {
0 } ) , 
RR* ,  <  ) )
4012, 39eqtrd 2470 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 370    = wceq 1437    e. wcel 1870   A.wral 2782    u. cun 3440   {csn 4002    |-> cmpt 4484   ran crn 4855    Fn wfn 5596   ` cfv 5601  (class class class)co 6305   supcsup 7960   0cc0 9538   RR*cxr 9673    < clt 9674   Basecbs 15084   distcds 15161   X_scprds 15303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-hom 15176  df-cco 15177  df-prds 15305
This theorem is referenced by:  prdsdsval3  15342  ressprdsds  21317
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