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Theorem ovolval 21755
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 9583 . . 3  |-  RR  e.  _V
21elpw2 4598 . 2  |-  ( A  e.  ~P RR  <->  A  C_  RR )
3 sseq1 3508 . . . . . . . 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 2952 . . . . . 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 3085 . . . . 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 2500 . . . 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 7905 . . 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 21746 . . 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 11353 . . . . 5  |-  <  Or  RR*
12 cnvso 5533 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1311, 12mpbi 208 . . . 4  |-  `'  <  Or 
RR*
1413supex 7922 . . 3  |-  sup ( M ,  RR* ,  `'  <  )  e.  _V
159, 10, 14fvmpt 5938 . 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 1381    e. wcel 1802   E.wrex 2792   {crab 2795    i^i cin 3458    C_ wss 3459   ~Pcpw 3994   U.cuni 4231    Or wor 4786    X. cxp 4984   `'ccnv 4985   ran crn 4987    o. ccom 4990   ` cfv 5575  (class class class)co 6278    ^m cmap 7419   supcsup 7899   RRcr 9491   1c1 9493    + caddc 9495   RR*cxr 9627    < clt 9628    <_ cle 9629    - cmin 9807   NNcn 10539   (,)cioo 11535    seqcseq 12083   abscabs 13043   vol*covol 21744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-pre-lttri 9566  ax-pre-lttrn 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-po 4787  df-so 4788  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-sup 7900  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-ovol 21746
This theorem is referenced by:  ovolcl  21759  ovollb  21760  ovolgelb  21761  ovolge0  21762  ovolsslem  21765  ovolshft  21792  ovolicc2  21803  ismblfin  30027
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