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Theorem ovolval 21648
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 4611 . 2  |-  ( A  e.  ~P RR  <->  A  C_  RR )
3 sseq1 3525 . . . . . . . 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 2973 . . . . . 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 3105 . . . . 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 2526 . . . 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 7906 . . 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 21639 . . 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 11347 . . . . 5  |-  <  Or  RR*
12 cnvso 5546 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1311, 12mpbi 208 . . . 4  |-  `'  <  Or 
RR*
1413supex 7923 . . 3  |-  sup ( M ,  RR* ,  `'  <  )  e.  _V
159, 10, 14fvmpt 5950 . 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 1379    e. wcel 1767   E.wrex 2815   {crab 2818    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   U.cuni 4245    Or wor 4799    X. cxp 4997   `'ccnv 4998   ran crn 5000    o. ccom 5003   ` cfv 5588  (class class class)co 6284    ^m cmap 7420   supcsup 7900   RRcr 9491   1c1 9493    + caddc 9495   RR*cxr 9627    < clt 9628    <_ cle 9629    - cmin 9805   NNcn 10536   (,)cioo 11529    seqcseq 12075   abscabs 13030   vol*covol 21637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  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 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-ovol 21639
This theorem is referenced by:  ovolcl  21652  ovollb  21653  ovolgelb  21654  ovolge0  21655  ovolsslem  21658  ovolshft  21685  ovolicc2  21696  ismblfin  29660
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