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Theorem elovolmr 20859
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 9369 . . . . . 6  |-  RR  e.  _V
21, 1xpex 6507 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
32inex2 4431 . . . 4  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
4 nnex 10324 . . . 4  |-  NN  e.  _V
53, 4elmap 7237 . . 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 22 . . . . . . . . . . . 12  |-  ( f  =  F  ->  f  =  F )
87eqcomd 2446 . . . . . . . . . . 11  |-  ( f  =  F  ->  F  =  f )
98coeq2d 4998 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  f )
)
109seqeq3d 11810 . . . . . . . . 9  |-  ( f  =  F  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  F
) )  =  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) )
116, 10syl5eq 2485 . . . . . . . 8  |-  ( f  =  F  ->  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) )
1211rneqd 5063 . . . . . . 7  |-  ( f  =  F  ->  ran  S  =  ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) )
1312supeq1d 7692 . . . . . 6  |-  ( f  =  F  ->  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) )
1413biantrud 504 . . . . 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 4994 . . . . . . . 8  |-  ( f  =  F  ->  ( (,)  o.  f )  =  ( (,)  o.  F
) )
1615rneqd 5063 . . . . . . 7  |-  ( f  =  F  ->  ran  ( (,)  o.  f )  =  ran  ( (,) 
o.  F ) )
1716unieqd 4098 . . . . . 6  |-  ( f  =  F  ->  U. ran  ( (,)  o.  f )  =  U. ran  ( (,)  o.  F ) )
1817sseq2d 3381 . . . . 5  |-  ( f  =  F  ->  ( A  C_  U. ran  ( (,)  o.  f )  <->  A  C_  U. ran  ( (,)  o.  F ) ) )
1914, 18bitr3d 255 . . . 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 3070 . . 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 470 . 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 20858 . 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 212 1  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  sup ( ran  S ,  RR* ,  <  )  e.  M )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   E.wrex 2714   {crab 2717    i^i cin 3324    C_ wss 3325   U.cuni 4088    X. cxp 4834   ran crn 4837    o. ccom 4840   -->wf 5411  (class class class)co 6090    ^m cmap 7210   supcsup 7686   RRcr 9277   1c1 9279    + caddc 9281   RR*cxr 9413    < clt 9414    <_ cle 9415    - cmin 9591   NNcn 10318   (,)cioo 11296    seqcseq 11802   abscabs 12719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-ico 11302  df-fz 11434  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721
This theorem is referenced by:  ovollb  20862  ovolshftlem1  20892
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