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Theorem ovolshftlem2 22006
Description: Lemma for ovolshft 22007. (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 727 . . . . . . 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 727 . . . . . . 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 727 . . . . . . 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 2382 . . . . . . 7  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  g )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  g )
)
9 fveq2 5774 . . . . . . . . . . 11  |-  ( m  =  n  ->  (
g `  m )  =  ( g `  n ) )
109fveq2d 5778 . . . . . . . . . 10  |-  ( m  =  n  ->  ( 1st `  ( g `  m ) )  =  ( 1st `  (
g `  n )
) )
1110oveq1d 6211 . . . . . . . . 9  |-  ( m  =  n  ->  (
( 1st `  (
g `  m )
)  +  C )  =  ( ( 1st `  ( g `  n
) )  +  C
) )
129fveq2d 5778 . . . . . . . . . 10  |-  ( m  =  n  ->  ( 2nd `  ( g `  m ) )  =  ( 2nd `  (
g `  n )
) )
1312oveq1d 6211 . . . . . . . . 9  |-  ( m  =  n  ->  (
( 2nd `  (
g `  m )
)  +  C )  =  ( ( 2nd `  ( g `  n
) )  +  C
) )
1411, 13opeq12d 4139 . . . . . . . 8  |-  ( m  =  n  ->  <. (
( 1st `  (
g `  m )
)  +  C ) ,  ( ( 2nd `  ( g `  m
) )  +  C
) >.  =  <. (
( 1st `  (
g `  n )
)  +  C ) ,  ( ( 2nd `  ( g `  n
) )  +  C
) >. )
1514cbvmptv 4458 . . . . . . 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 753 . . . . . . . 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 9494 . . . . . . . . . . 11  |-  RR  e.  _V
1817, 17xpex 6503 . . . . . . . . . 10  |-  ( RR 
X.  RR )  e. 
_V
1918inex2 4507 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
20 nnex 10458 . . . . . . . . 9  |-  NN  e.  _V
2119, 20elmap 7366 . . . . . . . 8  |-  ( g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  g : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2216, 21sylib 196 . . . . . . 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 459 . . . . . . 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 22005 . . . . . 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 2465 . . . . . 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 601 . . . 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 2874 . . 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 2796 . 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 3491 . 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 212 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 367    = wceq 1399    e. wcel 1826   A.wral 2732   E.wrex 2733   {crab 2736    i^i cin 3388    C_ wss 3389   <.cop 3950   U.cuni 4163    |-> cmpt 4425    X. cxp 4911   ran crn 4914    o. ccom 4917   -->wf 5492   ` cfv 5496  (class class class)co 6196   1stc1st 6697   2ndc2nd 6698    ^m cmap 7338   supcsup 7815   RRcr 9402   1c1 9404    + caddc 9406   RR*cxr 9538    < clt 9539    <_ cle 9540    - cmin 9718   NNcn 10452   (,)cioo 11450    seqcseq 12010   abscabs 13069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-ioo 11454  df-ico 11456  df-fz 11594  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071
This theorem is referenced by:  ovolshft  22007
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