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Theorem imasdsval2 14454
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 14453 . 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 4999 . . . . . . . . 9  |-  ( T  o.  g )  =  ( ( E  |`  ( V  X.  V
) )  o.  g
)
13 ssrab2 3437 . . . . . . . . . . . 12  |-  { 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 3386 . . . . . . . . . . 11  |-  S  C_  ( ( V  X.  V )  ^m  (
1 ... n ) )
1514sseli 3352 . . . . . . . . . 10  |-  ( g  e.  S  ->  g  e.  ( ( V  X.  V )  ^m  (
1 ... n ) ) )
16 elmapi 7234 . . . . . . . . . 10  |-  ( g  e.  ( ( V  X.  V )  ^m  ( 1 ... n
) )  ->  g : ( 1 ... n ) --> ( V  X.  V ) )
17 frn 5565 . . . . . . . . . 10  |-  ( g : ( 1 ... n ) --> ( V  X.  V )  ->  ran  g  C_  ( V  X.  V ) )
18 cores 5341 . . . . . . . . . 10  |-  ( ran  g  C_  ( V  X.  V )  ->  (
( E  |`  ( V  X.  V ) )  o.  g )  =  ( E  o.  g
) )
1915, 16, 17, 184syl 21 . . . . . . . . 9  |-  ( g  e.  S  ->  (
( E  |`  ( V  X.  V ) )  o.  g )  =  ( E  o.  g
) )
2012, 19syl5eq 2487 . . . . . . . 8  |-  ( g  e.  S  ->  ( T  o.  g )  =  ( E  o.  g ) )
2120oveq2d 6107 . . . . . . 7  |-  ( g  e.  S  ->  ( RR*s  gsumg  ( T  o.  g
) )  =  (
RR*s  gsumg  ( E  o.  g
) ) )
2221mpteq2ia 4374 . . . . . 6  |-  ( g  e.  S  |->  ( RR*s  gsumg  ( T  o.  g
) ) )  =  ( g  e.  S  |->  ( RR*s  gsumg  ( E  o.  g ) ) )
2322rneqi 5066 . . . . 5  |-  ran  (
g  e.  S  |->  (
RR*s  gsumg  ( T  o.  g
) ) )  =  ran  ( g  e.  S  |->  ( RR*s  gsumg  ( E  o.  g ) ) )
2423a1i 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
) ) ) )
2524iuneq2i 4189 . . 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 ) ) )
2625supeq1i 7697 . 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* ,  `'  <  )
2710, 26syl6eqr 2493 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719    C_ wss 3328   U_ciun 4171    e. cmpt 4350    X. cxp 4838   `'ccnv 4839   ran crn 4841    |` cres 4842    o. ccom 4844   -->wf 5414   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576    ^m cmap 7214   supcsup 7690   1c1 9283    + caddc 9285   RR*cxr 9417    < clt 9418    - cmin 9595   NNcn 10322   ...cfz 11437   Basecbs 14174   distcds 14247    gsumg cgsu 14379   RR*scxrs 14438    "s cimas 14442
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-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-imas 14446
This theorem is referenced by:  imasdsf1olem  19948
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