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Theorem ovolval 20955
Description: The value of 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
ovolval  |-  ( A 
C_  RR  ->  ( vol* `  A )  =  sup ( M ,  RR* ,  `'  <  )
)
Distinct variable group:    y, f, A
Allowed substitution hints:    M( y, f)

Proof of Theorem ovolval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 reex 9371 . . 3  |-  RR  e.  _V
21elpw2 4454 . 2  |-  ( A  e.  ~P RR  <->  A  C_  RR )
3 sseq1 3375 . . . . . . . 8  |-  ( x  =  A  ->  (
x  C_  U. ran  ( (,)  o.  f )  <->  A  C_  U. ran  ( (,)  o.  f ) ) )
43anbi1d 704 . . . . . . 7  |-  ( x  =  A  ->  (
( x  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) )  <-> 
( A  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) ) )
54rexbidv 2734 . . . . . 6  |-  ( x  =  A  ->  ( E. f  e.  (
(  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( x  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 )  /\  y  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) ) )
65rabbidv 2962 . . . . 5  |-  ( x  =  A  ->  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( x  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) }  =  { 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* ,  <  ) ) } )
7 ovolval.1 . . . . 5  |-  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* ,  <  ) ) }
86, 7syl6eqr 2491 . . . 4  |-  ( x  =  A  ->  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( x  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) }  =  M )
98supeq1d 7694 . . 3  |-  ( x  =  A  ->  sup ( { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( x  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) } ,  RR* ,  `'  <  )  =  sup ( M ,  RR* ,  `'  <  ) )
10 df-ovol 20946 . . 3  |-  vol* 
=  ( x  e. 
~P RR  |->  sup ( { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( x  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) } ,  RR* ,  `'  <  ) )
11 xrltso 11116 . . . . 5  |-  <  Or  RR*
12 cnvso 5374 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1311, 12mpbi 208 . . . 4  |-  `'  <  Or 
RR*
1413supex 7711 . . 3  |-  sup ( M ,  RR* ,  `'  <  )  e.  _V
159, 10, 14fvmpt 5772 . 2  |-  ( A  e.  ~P RR  ->  ( vol* `  A
)  =  sup ( M ,  RR* ,  `'  <  ) )
162, 15sylbir 213 1  |-  ( A 
C_  RR  ->  ( vol* `  A )  =  sup ( M ,  RR* ,  `'  <  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2714   {crab 2717    i^i cin 3325    C_ wss 3326   ~Pcpw 3858   U.cuni 4089    Or wor 4638    X. cxp 4836   `'ccnv 4837   ran crn 4839    o. ccom 4842   ` cfv 5416  (class class class)co 6089    ^m cmap 7212   supcsup 7688   RRcr 9279   1c1 9281    + caddc 9283   RR*cxr 9415    < clt 9416    <_ cle 9417    - cmin 9593   NNcn 10320   (,)cioo 11298    seqcseq 11804   abscabs 12721   vol*covol 20944
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-pre-lttri 9354  ax-pre-lttrn 9355
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-ovol 20946
This theorem is referenced by:  ovolcl  20959  ovollb  20960  ovolgelb  20961  ovolge0  20962  ovolsslem  20965  ovolshft  20992  ovolicc2  21003  ismblfin  28429
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