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Theorem imasdsval2 13734
Description: The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
imasbas.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasbas.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasbas.f  |-  ( ph  ->  F : V -onto-> B
)
imasbas.r  |-  ( ph  ->  R  e.  Z )
imasds.e  |-  E  =  ( dist `  R
)
imasds.d  |-  D  =  ( dist `  U
)
imasdsval.x  |-  ( ph  ->  X  e.  B )
imasdsval.y  |-  ( ph  ->  Y  e.  B )
imasdsval.s  |-  S  =  { h  e.  ( ( V  X.  V
)  ^m  ( 1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  X  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  Y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }
imasds.u  |-  T  =  ( E  |`  ( V  X.  V ) )
Assertion
Ref Expression
imasdsval2  |-  ( ph  ->  ( X D Y )  =  sup ( U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR* s  gsumg  ( T  o.  g ) ) ) ,  RR* ,  `'  <  ) )
Distinct variable groups:    g, h, i, n, F    R, g, h, i, n    ph, g, h, i, n    h, X, n    S, g    g, V, h    h, Y, n
Allowed substitution hints:    B( g, h, i, n)    D( g, h, i, n)    S( h, i, n)    T( g, h, i, n)    U( g, h, i, n)    E( g, h, i, n)    V( i, n)    X( g, i)    Y( g, i)    Z( g, h, i, n)

Proof of Theorem imasdsval2
StepHypRef Expression
1 imasbas.u . . 3  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasbas.v . . 3  |-  ( ph  ->  V  =  ( Base `  R ) )
3 imasbas.f . . 3  |-  ( ph  ->  F : V -onto-> B
)
4 imasbas.r . . 3  |-  ( ph  ->  R  e.  Z )
5 imasds.e . . 3  |-  E  =  ( dist `  R
)
6 imasds.d . . 3  |-  D  =  ( dist `  U
)
7 imasdsval.x . . 3  |-  ( ph  ->  X  e.  B )
8 imasdsval.y . . 3  |-  ( ph  ->  Y  e.  B )
9 imasdsval.s . . 3  |-  S  =  { h  e.  ( ( V  X.  V
)  ^m  ( 1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  X  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  Y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }
101, 2, 3, 4, 5, 6, 7, 8, 9imasdsval 13733 . 2  |-  ( ph  ->  ( X D Y )  =  sup ( U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR* s  gsumg  ( E  o.  g ) ) ) ,  RR* ,  `'  <  ) )
11 imasds.u . . . . . . . . . 10  |-  T  =  ( E  |`  ( V  X.  V ) )
1211coeq1i 5024 . . . . . . . . 9  |-  ( T  o.  g )  =  ( ( E  |`  ( V  X.  V
) )  o.  g
)
13 ssrab2 3420 . . . . . . . . . . . . 13  |-  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  X  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  Y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  C_  (
( V  X.  V
)  ^m  ( 1 ... n ) )
149, 13eqsstri 3370 . . . . . . . . . . . 12  |-  S  C_  ( ( V  X.  V )  ^m  (
1 ... n ) )
1514sseli 3336 . . . . . . . . . . 11  |-  ( g  e.  S  ->  g  e.  ( ( V  X.  V )  ^m  (
1 ... n ) ) )
16 elmapi 7030 . . . . . . . . . . 11  |-  ( g  e.  ( ( V  X.  V )  ^m  ( 1 ... n
) )  ->  g : ( 1 ... n ) --> ( V  X.  V ) )
1715, 16syl 16 . . . . . . . . . 10  |-  ( g  e.  S  ->  g : ( 1 ... n ) --> ( V  X.  V ) )
18 frn 5589 . . . . . . . . . 10  |-  ( g : ( 1 ... n ) --> ( V  X.  V )  ->  ran  g  C_  ( V  X.  V ) )
19 cores 5365 . . . . . . . . . 10  |-  ( ran  g  C_  ( V  X.  V )  ->  (
( E  |`  ( V  X.  V ) )  o.  g )  =  ( E  o.  g
) )
2017, 18, 193syl 19 . . . . . . . . 9  |-  ( g  e.  S  ->  (
( E  |`  ( V  X.  V ) )  o.  g )  =  ( E  o.  g
) )
2112, 20syl5eq 2479 . . . . . . . 8  |-  ( g  e.  S  ->  ( T  o.  g )  =  ( E  o.  g ) )
2221oveq2d 6089 . . . . . . 7  |-  ( g  e.  S  ->  ( RR* s  gsumg  ( T  o.  g
) )  =  (
RR* s  gsumg  ( E  o.  g
) ) )
2322mpteq2ia 4283 . . . . . 6  |-  ( g  e.  S  |->  ( RR* s  gsumg  ( T  o.  g
) ) )  =  ( g  e.  S  |->  ( RR* s  gsumg  ( E  o.  g ) ) )
2423rneqi 5088 . . . . 5  |-  ran  (
g  e.  S  |->  (
RR* s  gsumg  ( T  o.  g
) ) )  =  ran  ( g  e.  S  |->  ( RR* s  gsumg  ( E  o.  g ) ) )
2524a1i 11 . . . 4  |-  ( n  e.  NN  ->  ran  ( g  e.  S  |->  ( RR* s  gsumg  ( T  o.  g ) ) )  =  ran  (
g  e.  S  |->  (
RR* s  gsumg  ( E  o.  g
) ) ) )
2625iuneq2i 4103 . . 3  |-  U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR* s  gsumg  ( T  o.  g
) ) )  = 
U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR* s  gsumg  ( E  o.  g ) ) )
2726supeq1i 7444 . 2  |-  sup ( U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR* s  gsumg  ( T  o.  g ) ) ) ,  RR* ,  `'  <  )  =  sup ( U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR* s  gsumg  ( E  o.  g ) ) ) ,  RR* ,  `'  <  )
2810, 27syl6eqr 2485 1  |-  ( ph  ->  ( X D Y )  =  sup ( U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR* s  gsumg  ( T  o.  g ) ) ) ,  RR* ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    C_ wss 3312   U_ciun 4085    e. cmpt 4258    X. cxp 4868   `'ccnv 4869   ran crn 4871    |` cres 4872    o. ccom 4874   -->wf 5442   -onto->wfo 5444   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340    ^m cmap 7010   supcsup 7437   1c1 8983    + caddc 8985   RR*cxr 9111    < clt 9112    - cmin 9283   NNcn 9992   ...cfz 11035   Basecbs 13461   distcds 13530   RR* scxrs 13714    gsumg cgsu 13716    "s cimas 13722
This theorem is referenced by:  imasdsf1olem  18395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-imas 13726
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