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Theorem prdsdsval2 14728
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 2460 . . . . 5  |-  ( x  e.  I  |->  R )  =  ( x  e.  I  |->  R )
76fnmpt 5698 . . . 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 14722 . 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 2622 . . . . . . . 8  |-  F/_ x
( F `  y
)
14 nfcv 2622 . . . . . . . . 9  |-  F/_ x dist
15 nffvmpt1 5865 . . . . . . . . 9  |-  F/_ x
( ( x  e.  I  |->  R ) `  y )
1614, 15nffv 5864 . . . . . . . 8  |-  F/_ x
( dist `  ( (
x  e.  I  |->  R ) `  y ) )
17 nfcv 2622 . . . . . . . 8  |-  F/_ x
( G `  y
)
1813, 16, 17nfov 6298 . . . . . . 7  |-  F/_ x
( ( F `  y ) ( dist `  ( ( x  e.  I  |->  R ) `  y ) ) ( G `  y ) )
19 nfcv 2622 . . . . . . 7  |-  F/_ y
( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) )
20 fveq2 5857 . . . . . . . . 9  |-  ( y  =  x  ->  (
( x  e.  I  |->  R ) `  y
)  =  ( ( x  e.  I  |->  R ) `  x ) )
2120fveq2d 5861 . . . . . . . 8  |-  ( y  =  x  ->  ( dist `  ( ( x  e.  I  |->  R ) `
 y ) )  =  ( dist `  (
( x  e.  I  |->  R ) `  x
) ) )
22 fveq2 5857 . . . . . . . 8  |-  ( y  =  x  ->  ( F `  y )  =  ( F `  x ) )
23 fveq2 5857 . . . . . . . 8  |-  ( y  =  x  ->  ( G `  y )  =  ( G `  x ) )
2421, 22, 23oveq123d 6296 . . . . . . 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 4530 . . . . . 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 2461 . . . . . . 7  |-  ( ph  ->  I  =  I )
276fvmpt2 5948 . . . . . . . . . . . 12  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( ( x  e.  I  |->  R ) `  x )  =  R )
2827fveq2d 5861 . . . . . . . . . . 11  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( dist `  (
( x  e.  I  |->  R ) `  x
) )  =  (
dist `  R )
)
29 prdsdsval2.e . . . . . . . . . . 11  |-  E  =  ( dist `  R
)
3028, 29syl6eqr 2519 . . . . . . . . . 10  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( dist `  (
( x  e.  I  |->  R ) `  x
) )  =  E )
3130oveqd 6292 . . . . . . . . 9  |-  ( ( x  e.  I  /\  R  e.  X )  ->  ( ( F `  x ) ( dist `  ( ( x  e.  I  |->  R ) `  x ) ) ( G `  x ) )  =  ( ( F `  x ) E ( G `  x ) ) )
3231ralimiaa 2849 . . . . . . . 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 4519 . . . . . . 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 2513 . . . . 5  |-  ( ph  ->  ( y  e.  I  |->  ( ( F `  y ) ( dist `  ( ( x  e.  I  |->  R ) `  y ) ) ( G `  y ) ) )  =  ( x  e.  I  |->  ( ( F `  x
) E ( G `
 x ) ) ) )
3736rneqd 5221 . . . 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 3650 . . 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 7895 . 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 2501 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 1374    e. wcel 1762   A.wral 2807    u. cun 3467   {csn 4020    |-> cmpt 4498   ran crn 4993    Fn wfn 5574   ` cfv 5579  (class class class)co 6275   supcsup 7889   0cc0 9481   RR*cxr 9616    < clt 9617   Basecbs 14479   distcds 14553   X_scprds 14690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-hom 14568  df-cco 14569  df-prds 14692
This theorem is referenced by:  prdsdsval3  14729  ressprdsds  20602
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