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Theorem ovolshftlem2 22541
Description: Lemma for ovolshft 22542. (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 744 . . . . . . 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 744 . . . . . . 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 744 . . . . . . 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 2471 . . . . . . 7  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  g )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  g )
)
9 fveq2 5879 . . . . . . . . . . 11  |-  ( m  =  n  ->  (
g `  m )  =  ( g `  n ) )
109fveq2d 5883 . . . . . . . . . 10  |-  ( m  =  n  ->  ( 1st `  ( g `  m ) )  =  ( 1st `  (
g `  n )
) )
1110oveq1d 6323 . . . . . . . . 9  |-  ( m  =  n  ->  (
( 1st `  (
g `  m )
)  +  C )  =  ( ( 1st `  ( g `  n
) )  +  C
) )
129fveq2d 5883 . . . . . . . . . 10  |-  ( m  =  n  ->  ( 2nd `  ( g `  m ) )  =  ( 2nd `  (
g `  n )
) )
1312oveq1d 6323 . . . . . . . . 9  |-  ( m  =  n  ->  (
( 2nd `  (
g `  m )
)  +  C )  =  ( ( 2nd `  ( g `  n
) )  +  C
) )
1411, 13opeq12d 4166 . . . . . . . 8  |-  ( m  =  n  ->  <. (
( 1st `  (
g `  m )
)  +  C ) ,  ( ( 2nd `  ( g `  m
) )  +  C
) >.  =  <. (
( 1st `  (
g `  n )
)  +  C ) ,  ( ( 2nd `  ( g `  n
) )  +  C
) >. )
1514cbvmptv 4488 . . . . . . 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 770 . . . . . . . 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 9648 . . . . . . . . . . 11  |-  RR  e.  _V
1817, 17xpex 6614 . . . . . . . . . 10  |-  ( RR 
X.  RR )  e. 
_V
1918inex2 4538 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
20 nnex 10637 . . . . . . . . 9  |-  NN  e.  _V
2119, 20elmap 7518 . . . . . . . 8  |-  ( g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  g : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2216, 21sylib 201 . . . . . . 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 468 . . . . . . 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 22540 . . . . . 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 2544 . . . . . 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 17 . . . . 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 614 . . . 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 2871 . . 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 2809 . 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 3492 . 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 217 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 setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760    i^i cin 3389    C_ wss 3390   <.cop 3965   U.cuni 4190    |-> cmpt 4454    X. cxp 4837   ran crn 4840    o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811    ^m cmap 7490   supcsup 7972   RRcr 9556   1c1 9558    + caddc 9560   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   (,)cioo 11660    seqcseq 12251   abscabs 13374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-ioo 11664  df-ico 11666  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376
This theorem is referenced by:  ovolshft  22542
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