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Theorem imasdsval 14465
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 ) ) ) ) ) }
Assertion
Ref Expression
imasdsval  |-  ( ph  ->  ( X D Y )  =  sup ( U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR*s  gsumg  ( E  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)    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 imasdsval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
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
)
71, 2, 3, 4, 5, 6imasds 14463 . 2  |-  ( ph  ->  D  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  {
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 ) ) ) ) ) } 
|->  ( RR*s  gsumg  ( E  o.  g ) ) ) ,  RR* ,  `'  <  ) ) )
8 simplrl 759 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  x  =  X )
98eqeq2d 2454 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  ( ( F `  ( 1st `  ( h `  1
) ) )  =  x  <->  ( F `  ( 1st `  ( h `
 1 ) ) )  =  X ) )
10 simplrr 760 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  y  =  Y )
1110eqeq2d 2454 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  ( ( F `  ( 2nd `  ( h `  n
) ) )  =  y  <->  ( F `  ( 2nd `  ( h `
 n ) ) )  =  Y ) )
129, 113anbi12d 1290 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  ( ( ( 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 ) ) ) ) )  <->  ( ( 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 ) ) ) ) ) ) )
1312rabbidv 2976 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  { 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 ) ) ) ) ) }  =  {
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 ) ) ) ) ) } )
14 imasdsval.s . . . . . . 7  |-  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 ) ) ) ) ) }
1513, 14syl6eqr 2493 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  { 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 ) ) ) ) ) }  =  S )
1615mpteq1d 4385 . . . . 5  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  ( g  e.  { 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 ) ) ) ) ) }  |->  ( RR*s  gsumg  ( E  o.  g
) ) )  =  ( g  e.  S  |->  ( RR*s  gsumg  ( E  o.  g ) ) ) )
1716rneqd 5079 . . . 4  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  ran  (
g  e.  { 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 ) ) ) ) ) }  |->  ( RR*s  gsumg  ( E  o.  g
) ) )  =  ran  ( g  e.  S  |->  ( RR*s  gsumg  ( E  o.  g ) ) ) )
1817iuneq2dv 4204 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  U_ n  e.  NN  ran  ( g  e.  {
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 ) ) ) ) ) } 
|->  ( RR*s  gsumg  ( E  o.  g ) ) )  =  U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR*s  gsumg  ( E  o.  g
) ) ) )
1918supeq1d 7708 . 2  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  sup ( U_ n  e.  NN  ran  ( g  e.  { 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 ) ) ) ) ) }  |->  ( RR*s  gsumg  ( E  o.  g
) ) ) , 
RR* ,  `'  <  )  =  sup ( U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR*s  gsumg  ( E  o.  g ) ) ) ,  RR* ,  `'  <  ) )
20 imasdsval.x . 2  |-  ( ph  ->  X  e.  B )
21 imasdsval.y . 2  |-  ( ph  ->  Y  e.  B )
22 xrltso 11130 . . . . 5  |-  <  Or  RR*
23 cnvso 5388 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
2422, 23mpbi 208 . . . 4  |-  `'  <  Or 
RR*
2524supex 7725 . . 3  |-  sup ( U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR*s  gsumg  ( E  o.  g ) ) ) ,  RR* ,  `'  <  )  e.  _V
2625a1i 11 . 2  |-  ( ph  ->  sup ( U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR*s  gsumg  ( E  o.  g
) ) ) , 
RR* ,  `'  <  )  e.  _V )
277, 19, 20, 21, 26ovmpt2d 6230 1  |-  ( ph  ->  ( X D Y )  =  sup ( U_ n  e.  NN  ran  ( g  e.  S  |->  ( RR*s  gsumg  ( E  o.  g ) ) ) ,  RR* ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727   {crab 2731   _Vcvv 2984   U_ciun 4183    e. cmpt 4362    Or wor 4652    X. cxp 4850   `'ccnv 4851   ran crn 4853    o. ccom 4856   -onto->wfo 5428   ` cfv 5430  (class class class)co 6103   1stc1st 6587   2ndc2nd 6588    ^m cmap 7226   supcsup 7702   1c1 9295    + caddc 9297   RR*cxr 9429    < clt 9430    - cmin 9607   NNcn 10334   ...cfz 11449   Basecbs 14186   distcds 14259    gsumg cgsu 14391   RR*scxrs 14450    "s cimas 14454
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-fz 11450  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-plusg 14263  df-mulr 14264  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-imas 14458
This theorem is referenced by:  imasdsval2  14466
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