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Theorem ovolshft 18870
Description: The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.)
Hypotheses
Ref Expression
ovolshft.1  |-  ( ph  ->  A  C_  RR )
ovolshft.2  |-  ( ph  ->  C  e.  RR )
ovolshft.3  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
Assertion
Ref Expression
ovolshft  |-  ( ph  ->  ( vol * `  A )  =  ( vol * `  B
) )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem ovolshft
Dummy variables  f 
g  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovolshft.1 . . . . 5  |-  ( ph  ->  A  C_  RR )
2 ovolshft.2 . . . . 5  |-  ( ph  ->  C  e.  RR )
3 ovolshft.3 . . . . 5  |-  ( ph  ->  B  =  { x  e.  RR  |  ( x  -  C )  e.  A } )
4 eqid 2283 . . . . 5  |-  { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) ) }  =  { z  e. 
RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) ) }
51, 2, 3, 4ovolshftlem2 18869 . . . 4  |-  ( ph  ->  { 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* ,  <  ) ) }  C_  { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) ) } )
6 ssrab2 3258 . . . . . . 7  |-  { x  e.  RR  |  ( x  -  C )  e.  A }  C_  RR
76a1i 10 . . . . . 6  |-  ( ph  ->  { x  e.  RR  |  ( x  -  C )  e.  A }  C_  RR )
83, 7eqsstrd 3212 . . . . 5  |-  ( ph  ->  B  C_  RR )
92renegcld 9210 . . . . 5  |-  ( ph  -> 
-u C  e.  RR )
101, 2, 3shft2rab 18867 . . . . 5  |-  ( ph  ->  A  =  { w  e.  RR  |  ( w  -  -u C )  e.  B } )
11 eqid 2283 . . . . 5  |-  { 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* ,  <  ) ) }  =  { 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* ,  <  ) ) }
128, 9, 10, 11ovolshftlem2 18869 . . . 4  |-  ( ph  ->  { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U. ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  g ) ) , 
RR* ,  <  ) ) }  C_  { 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* ,  <  ) ) } )
135, 12eqssd 3196 . . 3  |-  ( ph  ->  { 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* ,  <  ) ) }  =  { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) ) } )
1413supeq1d 7199 . 2  |-  ( ph  ->  sup ( { 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* ,  <  ) ) } ,  RR* ,  `'  <  )  =  sup ( { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) ) } ,  RR* ,  `'  <  ) )
1511ovolval 18833 . . 3  |-  ( A 
C_  RR  ->  ( vol
* `  A )  =  sup ( { 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* ,  <  ) ) } ,  RR* ,  `'  <  ) )
161, 15syl 15 . 2  |-  ( ph  ->  ( vol * `  A )  =  sup ( { 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* ,  <  ) ) } ,  RR* ,  `'  <  ) )
174ovolval 18833 . . 3  |-  ( B 
C_  RR  ->  ( vol
* `  B )  =  sup ( { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U.
ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  g
) ) ,  RR* ,  <  ) ) } ,  RR* ,  `'  <  ) )
188, 17syl 15 . 2  |-  ( ph  ->  ( vol * `  B )  =  sup ( { z  e.  RR*  |  E. g  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( B  C_  U. ran  ( (,)  o.  g )  /\  z  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  g ) ) , 
RR* ,  <  ) ) } ,  RR* ,  `'  <  ) )
1914, 16, 183eqtr4d 2325 1  |-  ( ph  ->  ( vol * `  A )  =  ( vol * `  B
) )
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   U.cuni 3827    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   -ucneg 9038   NNcn 9746   (,)cioo 10656    seq cseq 11046   abscabs 11719   vol
*covol 18822
This theorem is referenced by:  shftmbl  18896  vitalilem4  18966
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-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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-reu 2550  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  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-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ioo 10660  df-ico 10662  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-ovol 18824
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