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Theorem shftmbl 18896
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 3258 . . 3  |-  { x  e.  RR  |  ( x  -  B )  e.  A }  C_  RR
21a1i 10 . 2  |-  ( ( A  e.  dom  vol  /\  B  e.  RR )  ->  { x  e.  RR  |  ( x  -  B )  e.  A }  C_  RR )
3 elpwi 3633 . . . 4  |-  ( y  e.  ~P RR  ->  y 
C_  RR )
4 simpll 730 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  A  e.  dom  vol )
5 ssrab2 3258 . . . . . . . 8  |-  { z  e.  RR  |  ( z  -  -u B
)  e.  y } 
C_  RR
65a1i 10 . . . . . . 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 732 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  y  C_  RR )
8 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  B  e.  RR )
98renegcld 9210 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  -u B  e.  RR )
10 eqidd 2284 . . . . . . . . 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 18870 . . . . . . . 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 733 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  ( vol * `  y )  e.  RR )
1311, 12eqeltrrd 2358 . . . . . . 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 18891 . . . . . . 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 1182 . . . . . 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 3389 . . . . . . . . 9  |-  ( y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } )  C_  y
1716, 7syl5ss 3190 . . . . . . . 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 18890 . . . . . . . . . . . 12  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
194, 18syl 15 . . . . . . . . . . 11  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol * `  y )  e.  RR ) )  ->  A  C_  RR )
20 eqidd 2284 . . . . . . . . . . 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 18867 . . . . . . . . . 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 3370 . . . . . . . . 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 3440 . . . . . . . . . 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 3358 . . . . . . . . . . . 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 10 . . . . . . . . . . 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 2778 . . . . . . . . . 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 2306 . . . . . . . . 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 2331 . . . . . . . 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 18870 . . . . . . 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
) ) )
30 difss 3303 . . . . . . . . 9  |-  ( y 
\  { x  e.  RR  |  ( x  -  B )  e.  A } )  C_  y
3130, 7syl5ss 3190 . . . . . . . 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 )
3221difeq2d 3294 . . . . . . . . 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 } } ) )
33 difrab 3442 . . . . . . . . . 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 } ) }
34 eldif 3162 . . . . . . . . . . . 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 } ) )
3534a1i 10 . . . . . . . . . . 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 } ) ) )
3635rabbiia 2778 . . . . . . . . . 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 } ) }
3733, 36eqtr4i 2306 . . . . . . . . 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 } ) }
3832, 37syl6eq 2331 . . . . . . . 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 }
) } )
3931, 9, 38ovolshft 18870 . . . . . . 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
) ) )
4029, 39oveq12d 5876 . . . . . 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 ) ) ) )
4115, 11, 403eqtr4d 2325 . . . . 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 }
) ) ) )
4241expr 598 . . . 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 } ) ) ) ) )
433, 42sylan2 460 . . 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 }
) ) ) ) )
4443ralrimiva 2626 . 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 } ) ) ) ) )
45 ismbl 18885 . 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 } ) ) ) ) ) )
462, 44, 45sylanbrc 645 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    \ cdif 3149    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   dom cdm 4689   ` cfv 5255  (class class class)co 5858   RRcr 8736    + caddc 8740    - cmin 9037   -ucneg 9038   vol *covol 18822   volcvol 18823
This theorem is referenced by:  vitalilem4  18966  vitalilem5  18967
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-icc 10663  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  df-vol 18825
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