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Theorem prdsdsval2 14422
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 2443 . . . . 5  |-  ( x  e.  I  |->  R )  =  ( x  e.  I  |->  R )
76fnmpt 5537 . . . 4  |-  ( A. x  e.  I  R  e.  X  ->  ( x  e.  I  |->  R )  Fn  I )
85, 7syl 16 . . 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 14416 . 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 2579 . . . . . . . 8  |-  F/_ x
( F `  y
)
14 nfcv 2579 . . . . . . . . 9  |-  F/_ x dist
15 nffvmpt1 5699 . . . . . . . . 9  |-  F/_ x
( ( x  e.  I  |->  R ) `  y )
1614, 15nffv 5698 . . . . . . . 8  |-  F/_ x
( dist `  ( (
x  e.  I  |->  R ) `  y ) )
17 nfcv 2579 . . . . . . . 8  |-  F/_ x
( G `  y
)
1813, 16, 17nfov 6114 . . . . . . 7  |-  F/_ x
( ( F `  y ) ( dist `  ( ( x  e.  I  |->  R ) `  y ) ) ( G `  y ) )
19 nfcv 2579 . . . . . . 7  |-  F/_ y
( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) )
20 fveq2 5691 . . . . . . . . 9  |-  ( y  =  x  ->  (
( x  e.  I  |->  R ) `  y
)  =  ( ( x  e.  I  |->  R ) `  x ) )
2120fveq2d 5695 . . . . . . . 8  |-  ( y  =  x  ->  ( dist `  ( ( x  e.  I  |->  R ) `
 y ) )  =  ( dist `  (
( x  e.  I  |->  R ) `  x
) ) )
22 fveq2 5691 . . . . . . . 8  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
23 fveq2 5691 . . . . . . . 8  |-  ( y  =  x  ->  ( G `  y )  =  ( G `  x ) )
2421, 22, 23oveq123d 6112 . . . . . . 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 4382 . . . . . 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 2444 . . . . . . 7  |-  ( ph  ->  I  =  I )
276fvmpt2 5781 . . . . . . . . . . . 12  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( ( x  e.  I  |->  R ) `  x )  =  R )
2827fveq2d 5695 . . . . . . . . . . 11  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( dist `  (
( x  e.  I  |->  R ) `  x
) )  =  (
dist `  R )
)
29 prdsdsval2.e . . . . . . . . . . 11  |-  E  =  ( dist `  R
)
3028, 29syl6eqr 2493 . . . . . . . . . 10  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( dist `  (
( x  e.  I  |->  R ) `  x
) )  =  E )
3130oveqd 6108 . . . . . . . . 9  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) )  =  ( ( F `  x ) E ( G `  x ) ) )
3231ralimiaa 2790 . . . . . . . 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 16 . . . . . . 7  |-  ( ph  ->  A. x  e.  I 
( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) )  =  ( ( F `  x ) E ( G `  x ) ) )
34 mpteq12 4371 . . . . . . 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 661 . . . . . 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 2487 . . . . 5  |-  ( ph  ->  ( y  e.  I  |->  ( ( F `  y ) ( dist `  ( ( x  e.  I  |->  R ) `  y ) ) ( G `  y ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) E ( G `
 x ) ) ) )
3736rneqd 5067 . . . 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 3509 . . 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 7696 . 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 2475 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 1369    e. wcel 1756   A.wral 2715    u. cun 3326   {csn 3877    e. cmpt 4350   ran crn 4841    Fn wfn 5413   ` cfv 5418  (class class class)co 6091   supcsup 7690   0cc0 9282   RR*cxr 9417    < clt 9418   Basecbs 14174   distcds 14247   X_scprds 14384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-hom 14262  df-cco 14263  df-prds 14386
This theorem is referenced by:  prdsdsval3  14423  ressprdsds  19946
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