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Theorem shftmbl 19433
Description: A shift of a measurable set is measurable. (Contributed by Mario Carneiro, 22-Mar-2014.)
Assertion
Ref Expression
shftmbl  |-  ( ( A  e.  dom  vol  /\  B  e.  RR )  ->  { x  e.  RR  |  ( x  -  B )  e.  A }  e.  dom  vol )
Distinct variable groups:    x, A    x, B

Proof of Theorem shftmbl
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3428 . . 3  |-  { x  e.  RR  |  ( x  -  B )  e.  A }  C_  RR
21a1i 11 . 2  |-  ( ( A  e.  dom  vol  /\  B  e.  RR )  ->  { x  e.  RR  |  ( x  -  B )  e.  A }  C_  RR )
3 elpwi 3807 . . . 4  |-  ( y  e.  ~P RR  ->  y 
C_  RR )
4 simpll 731 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  A  e.  dom  vol )
5 ssrab2 3428 . . . . . . . 8  |-  { z  e.  RR  |  ( z  -  -u B
)  e.  y } 
C_  RR
65a1i 11 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  { z  e.  RR  |  ( z  -  -u B
)  e.  y } 
C_  RR )
7 simprl 733 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  y  C_  RR )
8 simplr 732 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  B  e.  RR )
98renegcld 9464 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  -u B  e.  RR )
10 eqidd 2437 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  { z  e.  RR  |  ( z  -  -u B
)  e.  y }  =  { z  e.  RR  |  ( z  -  -u B )  e.  y } )
117, 9, 10ovolshft 19407 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( vol * `  y )  =  ( vol * `  { z  e.  RR  |  ( z  -  -u B )  e.  y } ) )
12 simprr 734 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( vol * `  y )  e.  RR )
1311, 12eqeltrrd 2511 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( vol * `  { z  e.  RR  |  ( z  -  -u B
)  e.  y } )  e.  RR )
14 mblsplit 19428 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\ 
{ z  e.  RR  |  ( z  -  -u B )  e.  y }  C_  RR  /\  ( vol * `  { z  e.  RR  |  ( z  -  -u B
)  e.  y } )  e.  RR )  ->  ( vol * `  { z  e.  RR  |  ( z  -  -u B )  e.  y } )  =  ( ( vol * `  ( { z  e.  RR  |  ( z  -  -u B )  e.  y }  i^i  A ) )  +  ( vol
* `  ( {
z  e.  RR  | 
( z  -  -u B
)  e.  y } 
\  A ) ) ) )
154, 6, 13, 14syl3anc 1184 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( vol * `  { z  e.  RR  |  ( z  -  -u B
)  e.  y } )  =  ( ( vol * `  ( { z  e.  RR  |  ( z  -  -u B )  e.  y }  i^i  A ) )  +  ( vol
* `  ( {
z  e.  RR  | 
( z  -  -u B
)  e.  y } 
\  A ) ) ) )
16 inss1 3561 . . . . . . . . 9  |-  ( y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } )  C_  y
1716, 7syl5ss 3359 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  (
y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } )  C_  RR )
18 mblss 19427 . . . . . . . . . . . 12  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
194, 18syl 16 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  A  C_  RR )
20 eqidd 2437 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  { x  e.  RR  |  ( x  -  B )  e.  A }  =  {
x  e.  RR  | 
( x  -  B
)  e.  A }
)
2119, 8, 20shft2rab 19404 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  A  =  { z  e.  RR  |  ( z  -  -u B )  e.  {
x  e.  RR  | 
( x  -  B
)  e.  A } } )
2221ineq2d 3542 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( { z  e.  RR  |  ( z  -  -u B )  e.  y }  i^i  A )  =  ( { z  e.  RR  |  ( z  -  -u B
)  e.  y }  i^i  { z  e.  RR  |  ( z  -  -u B )  e. 
{ x  e.  RR  |  ( x  -  B )  e.  A } } ) )
23 inrab 3613 . . . . . . . . . 10  |-  ( { z  e.  RR  | 
( z  -  -u B
)  e.  y }  i^i  { z  e.  RR  |  ( z  -  -u B )  e. 
{ x  e.  RR  |  ( x  -  B )  e.  A } } )  =  {
z  e.  RR  | 
( ( z  -  -u B )  e.  y  /\  ( z  -  -u B )  e.  {
x  e.  RR  | 
( x  -  B
)  e.  A }
) }
24 elin 3530 . . . . . . . . . . . 12  |-  ( ( z  -  -u B
)  e.  ( y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } )  <->  ( (
z  -  -u B
)  e.  y  /\  ( z  -  -u B
)  e.  { x  e.  RR  |  ( x  -  B )  e.  A } ) )
2524a1i 11 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  (
( z  -  -u B
)  e.  ( y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } )  <->  ( (
z  -  -u B
)  e.  y  /\  ( z  -  -u B
)  e.  { x  e.  RR  |  ( x  -  B )  e.  A } ) ) )
2625rabbiia 2946 . . . . . . . . . 10  |-  { z  e.  RR  |  ( z  -  -u B
)  e.  ( y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } ) }  =  { z  e.  RR  |  ( ( z  -  -u B
)  e.  y  /\  ( z  -  -u B
)  e.  { x  e.  RR  |  ( x  -  B )  e.  A } ) }
2723, 26eqtr4i 2459 . . . . . . . . 9  |-  ( { z  e.  RR  | 
( z  -  -u B
)  e.  y }  i^i  { z  e.  RR  |  ( z  -  -u B )  e. 
{ x  e.  RR  |  ( x  -  B )  e.  A } } )  =  {
z  e.  RR  | 
( z  -  -u B
)  e.  ( y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } ) }
2822, 27syl6eq 2484 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( { z  e.  RR  |  ( z  -  -u B )  e.  y }  i^i  A )  =  { z  e.  RR  |  ( z  -  -u B )  e.  ( y  i^i  {
x  e.  RR  | 
( x  -  B
)  e.  A }
) } )
2917, 9, 28ovolshft 19407 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( vol * `  ( y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } ) )  =  ( vol * `  ( { z  e.  RR  |  ( z  -  -u B )  e.  y }  i^i  A
) ) )
307ssdifssd 3485 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  (
y  \  { x  e.  RR  |  ( x  -  B )  e.  A } )  C_  RR )
3121difeq2d 3465 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( { z  e.  RR  |  ( z  -  -u B )  e.  y }  \  A )  =  ( { z  e.  RR  |  ( z  -  -u B
)  e.  y } 
\  { z  e.  RR  |  ( z  -  -u B )  e. 
{ x  e.  RR  |  ( x  -  B )  e.  A } } ) )
32 difrab 3615 . . . . . . . . . 10  |-  ( { z  e.  RR  | 
( z  -  -u B
)  e.  y } 
\  { z  e.  RR  |  ( z  -  -u B )  e. 
{ x  e.  RR  |  ( x  -  B )  e.  A } } )  =  {
z  e.  RR  | 
( ( z  -  -u B )  e.  y  /\  -.  ( z  -  -u B )  e. 
{ x  e.  RR  |  ( x  -  B )  e.  A } ) }
33 eldif 3330 . . . . . . . . . . . 12  |-  ( ( z  -  -u B
)  e.  ( y 
\  { x  e.  RR  |  ( x  -  B )  e.  A } )  <->  ( (
z  -  -u B
)  e.  y  /\  -.  ( z  -  -u B
)  e.  { x  e.  RR  |  ( x  -  B )  e.  A } ) )
3433a1i 11 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  (
( z  -  -u B
)  e.  ( y 
\  { x  e.  RR  |  ( x  -  B )  e.  A } )  <->  ( (
z  -  -u B
)  e.  y  /\  -.  ( z  -  -u B
)  e.  { x  e.  RR  |  ( x  -  B )  e.  A } ) ) )
3534rabbiia 2946 . . . . . . . . . 10  |-  { z  e.  RR  |  ( z  -  -u B
)  e.  ( y 
\  { x  e.  RR  |  ( x  -  B )  e.  A } ) }  =  { z  e.  RR  |  ( ( z  -  -u B
)  e.  y  /\  -.  ( z  -  -u B
)  e.  { x  e.  RR  |  ( x  -  B )  e.  A } ) }
3632, 35eqtr4i 2459 . . . . . . . . 9  |-  ( { z  e.  RR  | 
( z  -  -u B
)  e.  y } 
\  { z  e.  RR  |  ( z  -  -u B )  e. 
{ x  e.  RR  |  ( x  -  B )  e.  A } } )  =  {
z  e.  RR  | 
( z  -  -u B
)  e.  ( y 
\  { x  e.  RR  |  ( x  -  B )  e.  A } ) }
3731, 36syl6eq 2484 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( { z  e.  RR  |  ( z  -  -u B )  e.  y }  \  A )  =  { z  e.  RR  |  ( z  -  -u B )  e.  ( y  \  {
x  e.  RR  | 
( x  -  B
)  e.  A }
) } )
3830, 9, 37ovolshft 19407 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( vol * `  ( y 
\  { x  e.  RR  |  ( x  -  B )  e.  A } ) )  =  ( vol * `  ( { z  e.  RR  |  ( z  -  -u B )  e.  y }  \  A
) ) )
3929, 38oveq12d 6099 . . . . . 6  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  (
( vol * `  ( y  i^i  {
x  e.  RR  | 
( x  -  B
)  e.  A }
) )  +  ( vol * `  (
y  \  { x  e.  RR  |  ( x  -  B )  e.  A } ) ) )  =  ( ( vol * `  ( { z  e.  RR  |  ( z  -  -u B )  e.  y }  i^i  A ) )  +  ( vol
* `  ( {
z  e.  RR  | 
( z  -  -u B
)  e.  y } 
\  A ) ) ) )
4015, 11, 393eqtr4d 2478 . . . . 5  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( vol * `  y )  =  ( ( vol
* `  ( y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } ) )  +  ( vol * `  ( y  \  {
x  e.  RR  | 
( x  -  B
)  e.  A }
) ) ) )
4140expr 599 . . . 4  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  y  C_  RR )  ->  ( ( vol
* `  y )  e.  RR  ->  ( vol * `
 y )  =  ( ( vol * `  ( y  i^i  {
x  e.  RR  | 
( x  -  B
)  e.  A }
) )  +  ( vol * `  (
y  \  { x  e.  RR  |  ( x  -  B )  e.  A } ) ) ) ) )
423, 41sylan2 461 . . 3  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  y  e.  ~P RR )  ->  ( ( vol * `  y
)  e.  RR  ->  ( vol * `  y
)  =  ( ( vol * `  (
y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } ) )  +  ( vol * `  ( y  \  {
x  e.  RR  | 
( x  -  B
)  e.  A }
) ) ) ) )
4342ralrimiva 2789 . 2  |-  ( ( A  e.  dom  vol  /\  B  e.  RR )  ->  A. y  e.  ~P  RR ( ( vol * `  y )  e.  RR  ->  ( vol * `  y )  =  ( ( vol * `  ( y  i^i  {
x  e.  RR  | 
( x  -  B
)  e.  A }
) )  +  ( vol * `  (
y  \  { x  e.  RR  |  ( x  -  B )  e.  A } ) ) ) ) )
44 ismbl 19422 . 2  |-  ( { x  e.  RR  | 
( x  -  B
)  e.  A }  e.  dom  vol  <->  ( { x  e.  RR  |  ( x  -  B )  e.  A }  C_  RR  /\ 
A. y  e.  ~P  RR ( ( vol * `  y )  e.  RR  ->  ( vol * `  y )  =  ( ( vol * `  ( y  i^i  {
x  e.  RR  | 
( x  -  B
)  e.  A }
) )  +  ( vol * `  (
y  \  { x  e.  RR  |  ( x  -  B )  e.  A } ) ) ) ) ) )
452, 43, 44sylanbrc 646 1  |-  ( ( A  e.  dom  vol  /\  B  e.  RR )  ->  { x  e.  RR  |  ( x  -  B )  e.  A }  e.  dom  vol )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709    \ cdif 3317    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   dom cdm 4878   ` cfv 5454  (class class class)co 6081   RRcr 8989    + caddc 8993    - cmin 9291   -ucneg 9292   vol *covol 19359   volcvol 19360
This theorem is referenced by:  vitalilem4  19503  vitalilem5  19504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-ovol 19361  df-vol 19362
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