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Theorem ovolshftlem2 19274
Description: Lemma for ovolshft 19275. (Contributed by Mario Carneiro, 22-Mar-2014.)
Hypotheses
Ref Expression
ovolshft.1  |-  ( ph  ->  A  C_  RR )
ovolshft.2  |-  ( ph  ->  C  e.  RR )
ovolshft.3  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
ovolshft.4  |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
Assertion
Ref Expression
ovolshftlem2  |-  ( ph  ->  { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  g ) ) , 
RR* ,  <  ) ) }  C_  M )
Distinct variable groups:    f, g, x, y, z, A    C, f, g, x, y, z    B, f, g, y, z   
g, M, z    ph, f,
g, y, z
Allowed substitution hints:    ph( x)    B( x)    M( x, y, f)

Proof of Theorem ovolshftlem2
Dummy variables  n  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolshft.1 . . . . . . . 8  |-  ( ph  ->  A  C_  RR )
21ad3antrrr 711 . . . . . . 7  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  ->  A  C_  RR )
3 ovolshft.2 . . . . . . . 8  |-  ( ph  ->  C  e.  RR )
43ad3antrrr 711 . . . . . . 7  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  ->  C  e.  RR )
5 ovolshft.3 . . . . . . . 8  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
65ad3antrrr 711 . . . . . . 7  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
7 ovolshft.4 . . . . . . 7  |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
8 eqid 2388 . . . . . . 7  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  g )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  g )
)
9 fveq2 5669 . . . . . . . . . . 11  |-  ( m  =  n  ->  (
g `  m )  =  ( g `  n ) )
109fveq2d 5673 . . . . . . . . . 10  |-  ( m  =  n  ->  ( 1st `  ( g `  m ) )  =  ( 1st `  (
g `  n )
) )
1110oveq1d 6036 . . . . . . . . 9  |-  ( m  =  n  ->  (
( 1st `  (
g `  m )
)  +  C )  =  ( ( 1st `  ( g `  n
) )  +  C
) )
129fveq2d 5673 . . . . . . . . . 10  |-  ( m  =  n  ->  ( 2nd `  ( g `  m ) )  =  ( 2nd `  (
g `  n )
) )
1312oveq1d 6036 . . . . . . . . 9  |-  ( m  =  n  ->  (
( 2nd `  (
g `  m )
)  +  C )  =  ( ( 2nd `  ( g `  n
) )  +  C
) )
1411, 13opeq12d 3935 . . . . . . . 8  |-  ( m  =  n  ->  <. (
( 1st `  (
g `  m )
)  +  C ) ,  ( ( 2nd `  ( g `  m
) )  +  C
) >.  =  <. (
( 1st `  (
g `  n )
)  +  C ) ,  ( ( 2nd `  ( g `  n
) )  +  C
) >. )
1514cbvmptv 4242 . . . . . . 7  |-  ( m  e.  NN  |->  <. (
( 1st `  (
g `  m )
)  +  C ) ,  ( ( 2nd `  ( g `  m
) )  +  C
) >. )  =  ( n  e.  NN  |->  <.
( ( 1st `  (
g `  n )
)  +  C ) ,  ( ( 2nd `  ( g `  n
) )  +  C
) >. )
16 simplr 732 . . . . . . . 8  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  -> 
g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
17 reex 9015 . . . . . . . . . . 11  |-  RR  e.  _V
1817, 17xpex 4931 . . . . . . . . . 10  |-  ( RR 
X.  RR )  e. 
_V
1918inex2 4287 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
20 nnex 9939 . . . . . . . . 9  |-  NN  e.  _V
2119, 20elmap 6979 . . . . . . . 8  |-  ( g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  g : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2216, 21sylib 189 . . . . . . 7  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  -> 
g : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
23 simpr 448 . . . . . . 7  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  ->  A  C_  U. ran  ( (,)  o.  g ) )
242, 4, 6, 7, 8, 15, 22, 23ovolshftlem1 19273 . . . . . 6  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  ->  sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  g )
) ,  RR* ,  <  )  e.  M )
25 eleq1a 2457 . . . . . 6  |-  ( sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  g )
) ,  RR* ,  <  )  e.  M  ->  (
z  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  g ) ) , 
RR* ,  <  )  -> 
z  e.  M ) )
2624, 25syl 16 . . . . 5  |-  ( ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  /\  A  C_ 
U. ran  ( (,)  o.  g ) )  -> 
( z  =  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  g ) ) , 
RR* ,  <  )  -> 
z  e.  M ) )
2726expimpd 587 . . . 4  |-  ( ( ( ph  /\  z  e.  RR* )  /\  g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )  ->  (
( A  C_  U. ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  g ) ) , 
RR* ,  <  ) )  ->  z  e.  M
) )
2827rexlimdva 2774 . . 3  |-  ( (
ph  /\  z  e.  RR* )  ->  ( E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) )  -> 
z  e.  M ) )
2928ralrimiva 2733 . 2  |-  ( ph  ->  A. z  e.  RR*  ( E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  g ) ) , 
RR* ,  <  ) )  ->  z  e.  M
) )
30 rabss 3364 . 2  |-  ( { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) ) } 
C_  M  <->  A. z  e.  RR*  ( E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) )  -> 
z  e.  M ) )
3129, 30sylibr 204 1  |-  ( ph  ->  { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  g ) ) , 
RR* ,  <  ) ) }  C_  M )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650   E.wrex 2651   {crab 2654    i^i cin 3263    C_ wss 3264   <.cop 3761   U.cuni 3958    e. cmpt 4208    X. cxp 4817   ran crn 4820    o. ccom 4823   -->wf 5391   ` cfv 5395  (class class class)co 6021   1stc1st 6287   2ndc2nd 6288    ^m cmap 6955   supcsup 7381   RRcr 8923   1c1 8925    + caddc 8927   RR*cxr 9053    < clt 9054    <_ cle 9055    - cmin 9224   NNcn 9933   (,)cioo 10849    seq cseq 11251   abscabs 11967
This theorem is referenced by:  ovolshft  19275
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-ioo 10853  df-ico 10855  df-fz 10977  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969
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