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Theorem ovolshft 22371
Description: The Lebesgue outer measure function is shift-invariant. (Contributed by Mario Carneiro, 22-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
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 2420 . . . . 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 22370 . . . 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 3543 . . . . . 6  |-  { x  e.  RR  |  ( x  -  C )  e.  A }  C_  RR
73, 6syl6eqss 3511 . . . . 5  |-  ( ph  ->  B  C_  RR )
82renegcld 10035 . . . . 5  |-  ( ph  -> 
-u C  e.  RR )
91, 2, 3shft2rab 22368 . . . . 5  |-  ( ph  ->  A  =  { w  e.  RR  |  ( w  -  -u C )  e.  B } )
10 eqid 2420 . . . . 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 22370 . . . 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 3478 . . 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* ,  <  ) ) } )
1312infeq1d 7990 . 2  |-  ( ph  -> inf ( { 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* ,  <  )  = inf ( { 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 22333 . . 3  |-  ( A 
C_  RR  ->  ( vol* `  A )  = inf ( { 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 17 . 2  |-  ( ph  ->  ( vol* `  A )  = inf ( { 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 22333 . . 3  |-  ( B 
C_  RR  ->  ( vol* `  B )  = inf ( { 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 17 . 2  |-  ( ph  ->  ( vol* `  B )  = inf ( { 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 2471 1  |-  ( ph  ->  ( vol* `  A )  =  ( vol* `  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   E.wrex 2774   {crab 2777    i^i cin 3432    C_ wss 3433   U.cuni 4213    X. cxp 4843   ran crn 4846    o. ccom 4849   ` cfv 5592  (class class class)co 6296    ^m cmap 7471   supcsup 7951  infcinf 7952   RRcr 9527   1c1 9529    + caddc 9531   RR*cxr 9663    < clt 9664    <_ cle 9665    - cmin 9849   -ucneg 9850   NNcn 10598   (,)cioo 11624    seqcseq 12199   abscabs 13265   vol*covol 22320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-sup 7953  df-inf 7954  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-ioo 11628  df-ico 11630  df-fz 11772  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-ovol 22323
This theorem is referenced by:  shftmbl  22399  vitalilem4  22476
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