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Theorem ovolval 18849
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 8844 . . 3  |-  RR  e.  _V
21elpw2 4191 . 2  |-  ( A  e.  ~P RR  <->  A  C_  RR )
3 sseq1 3212 . . . . . . . 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 2577 . . . . . 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 2793 . . . . 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 2346 . . . 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 7215 . . 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 18840 . . 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 10491 . . . . 5  |-  <  Or  RR*
12 cnvso 5230 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1311, 12mpbi 199 . . . 4  |-  `'  <  Or 
RR*
1413supex 7230 . . 3  |-  sup ( M ,  RR* ,  `'  <  )  e.  _V
159, 10, 14fvmpt 5618 . 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 1632    e. wcel 1696   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843    Or wor 4329    X. cxp 4703   `'ccnv 4704   ran crn 4706    o. ccom 4709   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   supcsup 7209   RRcr 8752   1c1 8754    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   (,)cioo 10672    seq cseq 11062   abscabs 11735   vol
*covol 18838
This theorem is referenced by:  ovolcl  18853  ovollb  18854  ovolgelb  18855  ovolge0  18856  ovolsslem  18859  ovolshft  18886  ovolicc2  18897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-ovol 18840
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