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Theorem prdsdsval2 13399
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 2296 . . . . 5  |-  ( x  e.  I  |->  R )  =  ( x  e.  I  |->  R )
76fnmpt 5386 . . . 4  |-  ( A. x  e.  I  R  e.  X  ->  ( x  e.  I  |->  R )  Fn  I )
85, 7syl 15 . . 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 13393 . 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 2432 . . . . . . . 8  |-  F/_ x
( F `  y
)
14 nfcv 2432 . . . . . . . . 9  |-  F/_ x dist
15 nfmpt1 4125 . . . . . . . . . 10  |-  F/_ x
( x  e.  I  |->  R )
16 nfcv 2432 . . . . . . . . . 10  |-  F/_ x
y
1715, 16nffv 5548 . . . . . . . . 9  |-  F/_ x
( ( x  e.  I  |->  R ) `  y )
1814, 17nffv 5548 . . . . . . . 8  |-  F/_ x
( dist `  ( (
x  e.  I  |->  R ) `  y ) )
19 nfcv 2432 . . . . . . . 8  |-  F/_ x
( G `  y
)
2013, 18, 19nfov 5897 . . . . . . 7  |-  F/_ x
( ( F `  y ) ( dist `  ( ( x  e.  I  |->  R ) `  y ) ) ( G `  y ) )
21 nfcv 2432 . . . . . . 7  |-  F/_ y
( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) )
22 fveq2 5541 . . . . . . . . 9  |-  ( y  =  x  ->  (
( x  e.  I  |->  R ) `  y
)  =  ( ( x  e.  I  |->  R ) `  x ) )
2322fveq2d 5545 . . . . . . . 8  |-  ( y  =  x  ->  ( dist `  ( ( x  e.  I  |->  R ) `
 y ) )  =  ( dist `  (
( x  e.  I  |->  R ) `  x
) ) )
24 fveq2 5541 . . . . . . . 8  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
25 fveq2 5541 . . . . . . . 8  |-  ( y  =  x  ->  ( G `  y )  =  ( G `  x ) )
2623, 24, 25oveq123d 5895 . . . . . . 7  |-  ( y  =  x  ->  (
( F `  y
) ( dist `  (
( x  e.  I  |->  R ) `  y
) ) ( G `
 y ) )  =  ( ( F `
 x ) (
dist `  ( (
x  e.  I  |->  R ) `  x ) ) ( G `  x ) ) )
2720, 21, 26cbvmpt 4126 . . . . . 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 ) ) )
28 eqidd 2297 . . . . . . 7  |-  ( ph  ->  I  =  I )
296fvmpt2 5624 . . . . . . . . . . . 12  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( ( x  e.  I  |->  R ) `  x )  =  R )
3029fveq2d 5545 . . . . . . . . . . 11  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( dist `  (
( x  e.  I  |->  R ) `  x
) )  =  (
dist `  R )
)
31 prdsdsval2.e . . . . . . . . . . 11  |-  E  =  ( dist `  R
)
3230, 31syl6eqr 2346 . . . . . . . . . 10  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( dist `  (
( x  e.  I  |->  R ) `  x
) )  =  E )
3332oveqd 5891 . . . . . . . . 9  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) )  =  ( ( F `  x ) E ( G `  x ) ) )
3433ralimiaa 2630 . . . . . . . 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 ) ) )
355, 34syl 15 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) )  =  ( ( F `  x ) E ( G `  x ) ) )
36 mpteq12 4115 . . . . . . 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 ) ) ) )
3728, 35, 36syl2anc 642 . . . . . 6  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) E ( G `
 x ) ) ) )
3827, 37syl5eq 2340 . . . . 5  |-  ( ph  ->  ( y  e.  I  |->  ( ( F `  y ) ( dist `  ( ( x  e.  I  |->  R ) `  y ) ) ( G `  y ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) E ( G `
 x ) ) ) )
3938rneqd 4922 . . . 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 ) ) ) )
4039uneq1d 3341 . . 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 } ) )
4140supeq1d 7215 . 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* ,  <  ) )
4212, 41eqtrd 2328 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 358    = wceq 1632    e. wcel 1696   A.wral 2556    u. cun 3163   {csn 3653    e. cmpt 4093   ran crn 4706    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   supcsup 7209   0cc0 8753   RR*cxr 8882    < clt 8883   Basecbs 13164   distcds 13233   X_scprds 13362
This theorem is referenced by:  prdsdsval3  13400  ressprdsds  17951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364
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