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Theorem imasdsval 13741
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 13739 . 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 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  x  =  X )
98eqeq2d 2447 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  ( ( F `  ( 1st `  ( h `  1
) ) )  =  x  <->  ( F `  ( 1st `  ( h `
 1 ) ) )  =  X ) )
10 simplrr 738 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  y  =  Y )
1110eqeq2d 2447 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  X  /\  y  =  Y )
)  /\  n  e.  NN )  ->  ( ( F `  ( 2nd `  ( h `  n
) ) )  =  y  <->  ( F `  ( 2nd `  ( h `
 n ) ) )  =  Y ) )
129, 113anbi12d 1255 . . . . . . . 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 2948 . . . . . . 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 2486 . . . . . 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 4290 . . . . 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 5097 . . . 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 4114 . . 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 7451 . 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 10734 . . . . 5  |-  <  Or  RR*
23 cnvso 5411 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
2422, 23mpbi 200 . . . 4  |-  `'  <  Or 
RR*
2524supex 7468 . . 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 6201 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956   U_ciun 4093    e. cmpt 4266    Or wor 4502    X. cxp 4876   `'ccnv 4877   ran crn 4879    o. ccom 4882   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348    ^m cmap 7018   supcsup 7445   1c1 8991    + caddc 8993   RR*cxr 9119    < clt 9120    - cmin 9291   NNcn 10000   ...cfz 11043   Basecbs 13469   distcds 13538   RR* scxrs 13722    gsumg cgsu 13724    "s cimas 13730
This theorem is referenced by:  imasdsval2  13742
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-imas 13734
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