MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elovolm Structured version   Unicode version

Theorem elovolm 19373
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 2444 . . . . 5  |-  ( y  =  B  ->  (
y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  )  <->  B  =  sup ( ran  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
) ,  RR* ,  <  ) ) )
21anbi2d 686 . . . 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 2728 . . 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 3096 . 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 9083 . . . . . . . . . . . . 13  |-  RR  e.  _V
76, 6xpex 4992 . . . . . . . . . . . 12  |-  ( RR 
X.  RR )  e. 
_V
87inex2 4347 . . . . . . . . . . 11  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
9 nnex 10008 . . . . . . . . . . 11  |-  NN  e.  _V
108, 9elmap 7044 . . . . . . . . . 10  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
11 eqid 2438 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  o.  f )  =  ( ( abs  o.  -  )  o.  f )
12 eqid 2438 . . . . . . . . . . 11  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
1311, 12ovolsf 19371 . . . . . . . . . 10  |-  ( f : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> ( 0 [,)  +oo ) )
1410, 13sylbi 189 . . . . . . . . 9  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) : NN --> ( 0 [,) 
+oo ) )
15 icossxr 10997 . . . . . . . . 9  |-  ( 0 [,)  +oo )  C_  RR*
16 fss 5601 . . . . . . . . 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 645 . . . . . . . 8  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) : NN --> RR* )
18 frn 5599 . . . . . . . 8  |-  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) : NN --> RR* 
->  ran  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  f ) )  C_  RR* )
19 supxrcl 10895 . . . . . . . 8  |-  ( ran 
seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) )  C_  RR* 
->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  e.  RR* )
2017, 18, 193syl 19 . . . . . . 7  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  e.  RR* )
21 eleq1 2498 . . . . . . 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 215 . . . . . 6  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  ->  ( B  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  ->  B  e.  RR* ) )
2322imp 420 . . . . 5  |-  ( ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  B  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) )  ->  B  e.  RR* )
2423adantrl 698 . . . 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 2827 . . 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 616 . 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 245 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 set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   {crab 2711    i^i cin 3321    C_ wss 3322   U.cuni 4017    X. cxp 4878   ran crn 4881    o. ccom 4884   -->wf 5452  (class class class)co 6083    ^m cmap 7020   supcsup 7447   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    +oocpnf 9119   RR*cxr 9121    < clt 9122    <_ cle 9123    - cmin 9293   NNcn 10002   (,)cioo 10918   [,)cico 10920    seq cseq 11325   abscabs 12041
This theorem is referenced by:  elovolmr  19374  ovolmge0  19375  ovolicc2  19420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ico 10924  df-fz 11046  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043
  Copyright terms: Public domain W3C validator