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Theorem elovolmr 19325
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 9037 . . . . . 6  |-  RR  e.  _V
21, 1xpex 4949 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
32inex2 4305 . . . 4  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
4 nnex 9962 . . . 4  |-  NN  e.  _V
53, 4elmap 7001 . . 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 2409 . . . . . . . . . . 11  |-  ( f  =  F  ->  F  =  f )
98coeq2d 4994 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( abs  o.  -  )  o.  F )  =  ( ( abs  o.  -  )  o.  f )
)
109seqeq3d 11286 . . . . . . . . 9  |-  ( f  =  F  ->  seq  1 (  +  , 
( ( abs  o.  -  )  o.  F
) )  =  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) )
116, 10syl5eq 2448 . . . . . . . 8  |-  ( f  =  F  ->  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) )
1211rneqd 5056 . . . . . . 7  |-  ( f  =  F  ->  ran  S  =  ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) )
1312supeq1d 7409 . . . . . 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 4990 . . . . . . . 8  |-  ( f  =  F  ->  ( (,)  o.  f )  =  ( (,)  o.  F
) )
1615rneqd 5056 . . . . . . 7  |-  ( f  =  F  ->  ran  ( (,)  o.  f )  =  ran  ( (,) 
o.  F ) )
1716unieqd 3986 . . . . . 6  |-  ( f  =  F  ->  U. ran  ( (,)  o.  f )  =  U. ran  ( (,)  o.  F ) )
1817sseq2d 3336 . . . . 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 3012 . . 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 19324 . 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 1649    e. wcel 1721   E.wrex 2667   {crab 2670    i^i cin 3279    C_ wss 3280   U.cuni 3975    X. cxp 4835   ran crn 4838    o. ccom 4841   -->wf 5409  (class class class)co 6040    ^m cmap 6977   supcsup 7403   RRcr 8945   1c1 8947    + caddc 8949   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247   NNcn 9956   (,)cioo 10872    seq cseq 11278   abscabs 11994
This theorem is referenced by:  ovollb  19328  ovolshftlem1  19358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996
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