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Theorem elovolmr 19362
Description: Sufficient condition for elementhood in the set  M. (Contributed by Mario Carneiro, 16-Mar-2014.)
Hypotheses
Ref Expression
ovolval.1  |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
ovolval.2  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
Assertion
Ref Expression
elovolmr  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  sup ( ran  S ,  RR* ,  <  )  e.  M )
Distinct variable groups:    y, f, A    f, F    S, f,
y
Allowed substitution hints:    F( y)    M( y, f)

Proof of Theorem elovolmr
StepHypRef Expression
1 reex 9071 . . . . . 6  |-  RR  e.  _V
21, 1xpex 4982 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
32inex2 4337 . . . 4  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
4 nnex 9996 . . . 4  |-  NN  e.  _V
53, 4elmap 7034 . . 3  |-  ( F  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
6 ovolval.2 . . . . . . . . 9  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
7 id 20 . . . . . . . . . . . 12  |-  ( f  =  F  ->  f  =  F )
87eqcomd 2440 . . . . . . . . . . 11  |-  ( f  =  F  ->  F  =  f )
98coeq2d 5027 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  f )
)
109seqeq3d 11321 . . . . . . . . 9  |-  ( f  =  F  ->  seq  1 (  +  , 
( ( abs  o.  -  )  o.  F
) )  =  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) )
116, 10syl5eq 2479 . . . . . . . 8  |-  ( f  =  F  ->  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) )
1211rneqd 5089 . . . . . . 7  |-  ( f  =  F  ->  ran  S  =  ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) )
1312supeq1d 7443 . . . . . 6  |-  ( f  =  F  ->  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) )
1413biantrud 494 . . . . 5  |-  ( f  =  F  ->  ( A  C_  U. ran  ( (,)  o.  f )  <->  ( A  C_ 
U. ran  ( (,)  o.  f )  /\  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) ) )
15 coeq2 5023 . . . . . . . 8  |-  ( f  =  F  ->  ( (,)  o.  f )  =  ( (,)  o.  F
) )
1615rneqd 5089 . . . . . . 7  |-  ( f  =  F  ->  ran  ( (,)  o.  f )  =  ran  ( (,) 
o.  F ) )
1716unieqd 4018 . . . . . 6  |-  ( f  =  F  ->  U. ran  ( (,)  o.  f )  =  U. ran  ( (,)  o.  F ) )
1817sseq2d 3368 . . . . 5  |-  ( f  =  F  ->  ( A  C_  U. ran  ( (,)  o.  f )  <->  A  C_  U. ran  ( (,)  o.  F ) ) )
1914, 18bitr3d 247 . . . 4  |-  ( f  =  F  ->  (
( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran 
S ,  RR* ,  <  )  =  sup ( ran 
seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) )  <-> 
A  C_  U. ran  ( (,)  o.  F ) ) )
2019rspcev 3044 . . 3  |-  ( ( F  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  A  C_  U.
ran  ( (,)  o.  F ) )  ->  E. f  e.  (
(  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) )
215, 20sylanbr 460 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  E. f  e.  (
(  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) )
22 ovolval.1 . . 3  |-  M  =  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }
2322elovolm 19361 . 2  |-  ( sup ( ran  S ,  RR* ,  <  )  e.  M  <->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) )
2421, 23sylibr 204 1  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  sup ( ran  S ,  RR* ,  <  )  e.  M )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701    i^i cin 3311    C_ wss 3312   U.cuni 4007    X. cxp 4868   ran crn 4871    o. ccom 4874   -->wf 5442  (class class class)co 6073    ^m cmap 7010   supcsup 7437   RRcr 8979   1c1 8981    + caddc 8983   RR*cxr 9109    < clt 9110    <_ cle 9111    - cmin 9281   NNcn 9990   (,)cioo 10906    seq cseq 11313   abscabs 12029
This theorem is referenced by:  ovollb  19365  ovolshftlem1  19395
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-ico 10912  df-fz 11034  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031
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