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Theorem elovolmr 20959
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 9373 . . . . . 6  |-  RR  e.  _V
21, 1xpex 6508 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
32inex2 4434 . . . 4  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
4 nnex 10328 . . . 4  |-  NN  e.  _V
53, 4elmap 7241 . . 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 2448 . . . . . . . . . . 11  |-  ( f  =  F  ->  F  =  f )
98coeq2d 5002 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  f )
)
109seqeq3d 11814 . . . . . . . . 9  |-  ( f  =  F  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  F
) )  =  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) )
116, 10syl5eq 2487 . . . . . . . 8  |-  ( f  =  F  ->  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) )
1211rneqd 5067 . . . . . . 7  |-  ( f  =  F  ->  ran  S  =  ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) )
1312supeq1d 7696 . . . . . 6  |-  ( f  =  F  ->  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) )
1413biantrud 507 . . . . 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 4998 . . . . . . . 8  |-  ( f  =  F  ->  ( (,)  o.  f )  =  ( (,)  o.  F
) )
1615rneqd 5067 . . . . . . 7  |-  ( f  =  F  ->  ran  ( (,)  o.  f )  =  ran  ( (,) 
o.  F ) )
1716unieqd 4101 . . . . . 6  |-  ( f  =  F  ->  U. ran  ( (,)  o.  f )  =  U. ran  ( (,)  o.  F ) )
1817sseq2d 3384 . . . . 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 3073 . . 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 473 . 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 20958 . 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 1369    e. wcel 1756   E.wrex 2716   {crab 2719    i^i cin 3327    C_ wss 3328   U.cuni 4091    X. cxp 4838   ran crn 4841    o. ccom 4844   -->wf 5414  (class class class)co 6091    ^m cmap 7214   supcsup 7690   RRcr 9281   1c1 9283    + caddc 9285   RR*cxr 9417    < clt 9418    <_ cle 9419    - cmin 9595   NNcn 10322   (,)cioo 11300    seqcseq 11806   abscabs 12723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-ico 11306  df-fz 11438  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725
This theorem is referenced by:  ovollb  20962  ovolshftlem1  20992
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