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Theorem elovolm 22011
Description: Elementhood in the set  M of approximations to the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.)
Hypothesis
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* ,  <  ) ) }
Assertion
Ref Expression
elovolm  |-  ( B  e.  M  <->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) )
Distinct variable groups:    y, f, A    B, f, y
Allowed substitution hints:    M( y, f)

Proof of Theorem elovolm
StepHypRef Expression
1 eqeq1 2461 . . . . 5  |-  ( y  =  B  ->  (
y  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  )  <->  B  =  sup ( ran  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  ) ) )
21anbi2d 703 . . . 4  |-  ( y  =  B  ->  (
( A  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) )  <-> 
( A  C_  U. ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) ) )
32rexbidv 2968 . . 3  |-  ( y  =  B  ->  ( 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* ,  <  ) )  <->  E. f  e.  (
(  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) ) )
4 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* ,  <  ) ) }
53, 4elrab2 3259 . 2  |-  ( B  e.  M  <->  ( B  e.  RR*  /\  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) ) )
6 reex 9600 . . . . . . . . . . . . 13  |-  RR  e.  _V
76, 6xpex 6603 . . . . . . . . . . . 12  |-  ( RR 
X.  RR )  e. 
_V
87inex2 4598 . . . . . . . . . . 11  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
9 nnex 10562 . . . . . . . . . . 11  |-  NN  e.  _V
108, 9elmap 7466 . . . . . . . . . 10  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
11 eqid 2457 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  o.  f )  =  ( ( abs  o.  -  )  o.  f )
12 eqid 2457 . . . . . . . . . . 11  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
1311, 12ovolsf 22009 . . . . . . . . . 10  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,) +oo ) )
1410, 13sylbi 195 . . . . . . . . 9  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) : NN --> ( 0 [,) +oo ) )
15 icossxr 11634 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  RR*
16 fss 5745 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) : NN --> ( 0 [,) +oo )  /\  (
0 [,) +oo )  C_ 
RR* )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> RR* )
1714, 15, 16sylancl 662 . . . . . . . 8  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) : NN --> RR* )
18 frn 5743 . . . . . . . 8  |-  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> RR* 
->  ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) )  C_  RR* )
19 supxrcl 11531 . . . . . . . 8  |-  ( ran 
seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) )  C_  RR* 
->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  e.  RR* )
2017, 18, 193syl 20 . . . . . . 7  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  e.  RR* )
21 eleq1 2529 . . . . . . 7  |-  ( B  =  sup ( ran 
seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  )  -> 
( B  e.  RR*  <->  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  e. 
RR* ) )
2220, 21syl5ibrcom 222 . . . . . 6  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  ( B  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  ->  B  e.  RR* ) )
2322imp 429 . . . . 5  |-  ( ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  B  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) )  ->  B  e.  RR* )
2423adantrl 715 . . . 4  |-  ( ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( A  C_  U. ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) )  ->  B  e.  RR* )
2524rexlimiva 2945 . . 3  |-  ( E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) )  ->  B  e.  RR* )
2625pm4.71ri 633 . 2  |-  ( E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) )  <->  ( B  e.  RR*  /\  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) ) )
275, 26bitr4i 252 1  |-  ( B  e.  M  <->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  B  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   {crab 2811    i^i cin 3470    C_ wss 3471   U.cuni 4251    X. cxp 5006   ran crn 5009    o. ccom 5012   -->wf 5590  (class class class)co 6296    ^m cmap 7438   supcsup 7918   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824   NNcn 10556   (,)cioo 11554   [,)cico 11556    seqcseq 12109   abscabs 13078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ico 11560  df-fz 11698  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080
This theorem is referenced by:  elovolmr  22012  ovolmge0  22013  ovolicc2  22058
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