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Theorem ovolshft 19409
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 2438 . . . . 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 19408 . . . 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 3430 . . . . . 6  |-  { x  e.  RR  |  ( x  -  C )  e.  A }  C_  RR
73, 6syl6eqss 3400 . . . . 5  |-  ( ph  ->  B  C_  RR )
82renegcld 9466 . . . . 5  |-  ( ph  -> 
-u C  e.  RR )
91, 2, 3shft2rab 19406 . . . . 5  |-  ( ph  ->  A  =  { w  e.  RR  |  ( w  -  -u C )  e.  B } )
10 eqid 2438 . . . . 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* ,  <  ) ) }
117, 8, 9, 10ovolshftlem2 19408 . . . 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* ,  <  ) ) } )
125, 11eqssd 3367 . . 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* ,  <  ) ) } )
1312supeq1d 7453 . 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* ,  `'  <  ) )
1410ovolval 19372 . . 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* ,  `'  <  ) )
151, 14syl 16 . 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* ,  `'  <  ) )
164ovolval 19372 . . 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* ,  `'  <  ) )
177, 16syl 16 . 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* ,  `'  <  ) )
1813, 15, 173eqtr4d 2480 1  |-  ( ph  ->  ( vol * `  A )  =  ( vol * `  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   {crab 2711    i^i cin 3321    C_ wss 3322   U.cuni 4017    X. cxp 4878   `'ccnv 4879   ran crn 4881    o. ccom 4884   ` cfv 5456  (class class class)co 6083    ^m cmap 7020   supcsup 7447   RRcr 8991   1c1 8993    + caddc 8995   RR*cxr 9121    < clt 9122    <_ cle 9123    - cmin 9293   -ucneg 9294   NNcn 10002   (,)cioo 10918    seq cseq 11325   abscabs 12041   vol
*covol 19361
This theorem is referenced by:  shftmbl  19435  vitalilem4  19505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ioo 10922  df-ico 10924  df-fz 11046  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-ovol 19363
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