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Theorem shftmbl 22074
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 3581 . . 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 4024 . . . 4  |-  ( y  e.  ~P RR  ->  y 
C_  RR )
4 simpll 753 . . . . . . 7  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol* `  y )  e.  RR ) )  ->  A  e.  dom  vol )
5 ssrab2 3581 . . . . . . . 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 756 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol* `  y )  e.  RR ) )  ->  y  C_  RR )
8 simplr 755 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol* `  y )  e.  RR ) )  ->  B  e.  RR )
98renegcld 10007 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol* `  y )  e.  RR ) )  ->  -u B  e.  RR )
10 eqidd 2458 . . . . . . . . 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 22047 . . . . . . . 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 757 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol* `  y )  e.  RR ) )  ->  ( vol* `  y )  e.  RR )
1311, 12eqeltrrd 2546 . . . . . . 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 22068 . . . . . . 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 1228 . . . . . 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 3714 . . . . . . . . 9  |-  ( y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } )  C_  y
1716, 7syl5ss 3510 . . . . . . . 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 22067 . . . . . . . . . . . 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 2458 . . . . . . . . . . 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 22044 . . . . . . . . . 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 3696 . . . . . . . . 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 3777 . . . . . . . . . 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 3683 . . . . . . . . . . . 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 3098 . . . . . . . . . 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 2489 . . . . . . . . 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 2514 . . . . . . . 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 22047 . . . . . . 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 3638 . . . . . . . 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 3618 . . . . . . . . 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 3779 . . . . . . . . . 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 3481 . . . . . . . . . . . 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 3098 . . . . . . . . . 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 2489 . . . . . . . . 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 2514 . . . . . . . 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 22047 . . . . . . 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 6314 . . . . . 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 2508 . . . . 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 615 . . . 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 474 . . 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 2871 . 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 22062 . 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 664 1  |-  ( ( A  e.  dom  vol  /\  B  e.  RR )  ->  { x  e.  RR  |  ( x  -  B )  e.  A }  e.  dom  vol )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811    \ cdif 3468    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   dom cdm 5008   ` cfv 5594  (class class class)co 6296   RRcr 9508    + caddc 9512    - cmin 9824   -ucneg 9825   vol*covol 21999   volcvol 22000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-ovol 22001  df-vol 22002
This theorem is referenced by:  vitalilem4  22145  vitalilem5  22146
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