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Theorem imasdsval 13696
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 13694 . 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 2415 . . . . . . . . 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 2415 . . . . . . . . 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 2908 . . . . . . 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 2454 . . . . . 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 4250 . . . . 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 5056 . . . 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 4074 . . 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 7409 . 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 10690 . . . . 5  |-  <  Or  RR*
23 cnvso 5370 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
2422, 23mpbi 200 . . . 4  |-  `'  <  Or 
RR*
2524supex 7424 . . 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 6160 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 1649    e. wcel 1721   A.wral 2666   {crab 2670   _Vcvv 2916   U_ciun 4053    e. cmpt 4226    Or wor 4462    X. cxp 4835   `'ccnv 4836   ran crn 4838    o. ccom 4841   -onto->wfo 5411   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307    ^m cmap 6977   supcsup 7403   1c1 8947    + caddc 8949   RR*cxr 9075    < clt 9076    - cmin 9247   NNcn 9956   ...cfz 10999   Basecbs 13424   distcds 13493   RR* scxrs 13677    gsumg cgsu 13679    "s cimas 13685
This theorem is referenced by:  imasdsval2  13697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-imas 13689
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