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Theorem ovolval 18833
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 8828 . . 3  |-  RR  e.  _V
21elpw2 4175 . 2  |-  ( A  e.  ~P RR  <->  A  C_  RR )
3 sseq1 3199 . . . . . . . 8  |-  ( x  =  A  ->  (
x  C_  U. ran  ( (,)  o.  f )  <->  A  C_  U. ran  ( (,)  o.  f ) ) )
43anbi1d 685 . . . . . . 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 2564 . . . . . 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 2780 . . . . 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 2333 . . . 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 7199 . . 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 18824 . . 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 10475 . . . . 5  |-  <  Or  RR*
12 cnvso 5214 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1311, 12mpbi 199 . . . 4  |-  `'  <  Or 
RR*
1413supex 7214 . . 3  |-  sup ( M ,  RR* ,  `'  <  )  e.  _V
159, 10, 14fvmpt 5602 . 2  |-  ( A  e.  ~P RR  ->  ( vol * `  A
)  =  sup ( M ,  RR* ,  `'  <  ) )
162, 15sylbir 204 1  |-  ( A 
C_  RR  ->  ( vol
* `  A )  =  sup ( M ,  RR* ,  `'  <  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827    Or wor 4313    X. cxp 4687   `'ccnv 4688   ran crn 4690    o. ccom 4693   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   supcsup 7193   RRcr 8736   1c1 8738    + caddc 8740   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   (,)cioo 10656    seq cseq 11046   abscabs 11719   vol
*covol 18822
This theorem is referenced by:  ovolcl  18837  ovollb  18838  ovolgelb  18839  ovolge0  18840  ovolsslem  18843  ovolshft  18870  ovolicc2  18881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-ovol 18824
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