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Theorem shftmbl 21042
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 3458 . . 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 3890 . . . 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 3458 . . . . . . . 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 755 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol* `  y )  e.  RR ) )  ->  y  C_  RR )
8 simplr 754 . . . . . . . . . 10  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol* `  y )  e.  RR ) )  ->  B  e.  RR )
98renegcld 9796 . . . . . . . . 9  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol* `  y )  e.  RR ) )  ->  -u B  e.  RR )
10 eqidd 2444 . . . . . . . . 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 21016 . . . . . . . 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 756 . . . . . . . 8  |-  ( ( ( A  e.  dom  vol 
/\  B  e.  RR )  /\  ( y  C_  RR  /\  ( vol* `  y )  e.  RR ) )  ->  ( vol* `  y )  e.  RR )
1311, 12eqeltrrd 2518 . . . . . . 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 21037 . . . . . . 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 1218 . . . . . 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 3591 . . . . . . . . 9  |-  ( y  i^i  { x  e.  RR  |  ( x  -  B )  e.  A } )  C_  y
1716, 7syl5ss 3388 . . . . . . . 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 21036 . . . . . . . . . . . 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 2444 . . . . . . . . . . 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 21013 . . . . . . . . . 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 3573 . . . . . . . . 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 3643 . . . . . . . . . 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 3560 . . . . . . . . . . . 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 2982 . . . . . . . . . 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 2466 . . . . . . . . 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 2491 . . . . . . . 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 21016 . . . . . . 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 3515 . . . . . . . 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 3495 . . . . . . . . 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 3645 . . . . . . . . . 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 3359 . . . . . . . . . . . 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 2982 . . . . . . . . . 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 2466 . . . . . . . . 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 2491 . . . . . . . 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 21016 . . . . . . 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 6130 . . . . . 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 2485 . . . . 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 2820 . 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 21031 . 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 1369    e. wcel 1756   A.wral 2736   {crab 2740    \ cdif 3346    i^i cin 3348    C_ wss 3349   ~Pcpw 3881   dom cdm 4861   ` cfv 5439  (class class class)co 6112   RRcr 9302    + caddc 9306    - cmin 9616   -ucneg 9617   vol*covol 20968   volcvol 20969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-ioo 11325  df-ico 11327  df-icc 11328  df-fz 11459  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-ovol 20970  df-vol 20971
This theorem is referenced by:  vitalilem4  21113  vitalilem5  21114
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