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Theorem ioombl1 21065
Description: An open right-unbounded interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by Mario Carneiro, 25-Mar-2015.)
Assertion
Ref Expression
ioombl1  |-  ( A  e.  RR*  ->  ( A (,) +oo )  e. 
dom  vol )

Proof of Theorem ioombl1
Dummy variables  f  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxr 11117 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 ioossre 11378 . . . . 5  |-  ( A (,) +oo )  C_  RR
32a1i 11 . . . 4  |-  ( A  e.  RR  ->  ( A (,) +oo )  C_  RR )
4 elpwi 3890 . . . . . 6  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
5 simplrl 759 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  x  C_  RR )
6 simplrr 760 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  ( vol* `  x )  e.  RR )
7 simpr 461 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  y  e.  RR+ )
8 eqid 2443 . . . . . . . . . . . 12  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
98ovolgelb 20985 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR  /\  y  e.  RR+ )  ->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( x  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  +  y ) ) )
105, 6, 7, 9syl3anc 1218 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( x  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  +  y ) ) )
11 eqid 2443 . . . . . . . . . . 11  |-  ( A (,) +oo )  =  ( A (,) +oo )
12 simplll 757 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  /\  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( x  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  +  y ) ) ) )  ->  A  e.  RR )
135adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  /\  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( x  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  +  y ) ) ) )  ->  x  C_  RR )
146adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  /\  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( x  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  +  y ) ) ) )  ->  ( vol* `  x )  e.  RR )
15 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  /\  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( x  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  +  y ) ) ) )  ->  y  e.  RR+ )
16 eqid 2443 . . . . . . . . . . 11  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  ( m  e.  NN  |->  <. if ( if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  <_ 
( 2nd `  (
f `  m )
) ,  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) ) ,  ( 2nd `  (
f `  m )
) ) ,  ( 2nd `  ( f `
 m ) )
>. ) ) )  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  ( m  e.  NN  |->  <. if ( if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  <_ 
( 2nd `  (
f `  m )
) ,  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) ) ,  ( 2nd `  (
f `  m )
) ) ,  ( 2nd `  ( f `
 m ) )
>. ) ) )
17 eqid 2443 . . . . . . . . . . 11  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  ( m  e.  NN  |->  <. ( 1st `  (
f `  m )
) ,  if ( if ( ( 1st `  ( f `  m
) )  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  <_ 
( 2nd `  (
f `  m )
) ,  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) ) ,  ( 2nd `  (
f `  m )
) ) >. )
) )  =  seq 1 (  +  , 
( ( abs  o.  -  )  o.  (
m  e.  NN  |->  <.
( 1st `  (
f `  m )
) ,  if ( if ( ( 1st `  ( f `  m
) )  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  <_ 
( 2nd `  (
f `  m )
) ,  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) ) ,  ( 2nd `  (
f `  m )
) ) >. )
) )
18 simprl 755 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  /\  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( x  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  +  y ) ) ) )  ->  f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
19 reex 9394 . . . . . . . . . . . . . . 15  |-  RR  e.  _V
2019, 19xpex 6529 . . . . . . . . . . . . . 14  |-  ( RR 
X.  RR )  e. 
_V
2120inex2 4455 . . . . . . . . . . . . 13  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
22 nnex 10349 . . . . . . . . . . . . 13  |-  NN  e.  _V
2321, 22elmap 7262 . . . . . . . . . . . 12  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2418, 23sylib 196 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  /\  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( x  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  +  y ) ) ) )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
25 simprrl 763 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  /\  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( x  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  +  y ) ) ) )  ->  x  C_  U. ran  ( (,)  o.  f ) )
26 simprrr 764 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  /\  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( x  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  +  y ) ) ) )  ->  sup ( ran  seq 1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  )  <_  (
( vol* `  x )  +  y ) )
27 eqid 2443 . . . . . . . . . . 11  |-  ( 1st `  ( f `  n
) )  =  ( 1st `  ( f `
 n ) )
28 eqid 2443 . . . . . . . . . . 11  |-  ( 2nd `  ( f `  n
) )  =  ( 2nd `  ( f `
 n ) )
29 fveq2 5712 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  n  ->  (
f `  m )  =  ( f `  n ) )
3029fveq2d 5716 . . . . . . . . . . . . . . . . 17  |-  ( m  =  n  ->  ( 1st `  ( f `  m ) )  =  ( 1st `  (
f `  n )
) )
3130breq1d 4323 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
( 1st `  (
f `  m )
)  <_  A  <->  ( 1st `  ( f `  n
) )  <_  A
) )
3231, 30ifbieq2d 3835 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  =  if ( ( 1st `  ( f `  n
) )  <_  A ,  A ,  ( 1st `  ( f `  n
) ) ) )
3329fveq2d 5716 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  ( 2nd `  ( f `  m ) )  =  ( 2nd `  (
f `  n )
) )
3432, 33breq12d 4326 . . . . . . . . . . . . . 14  |-  ( m  =  n  ->  ( if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  <_ 
( 2nd `  (
f `  m )
)  <->  if ( ( 1st `  ( f `  n
) )  <_  A ,  A ,  ( 1st `  ( f `  n
) ) )  <_ 
( 2nd `  (
f `  n )
) ) )
3534, 32, 33ifbieq12d 3837 . . . . . . . . . . . . 13  |-  ( m  =  n  ->  if ( if ( ( 1st `  ( f `  m
) )  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  <_ 
( 2nd `  (
f `  m )
) ,  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) ) ,  ( 2nd `  (
f `  m )
) )  =  if ( if ( ( 1st `  ( f `
 n ) )  <_  A ,  A ,  ( 1st `  (
f `  n )
) )  <_  ( 2nd `  ( f `  n ) ) ,  if ( ( 1st `  ( f `  n
) )  <_  A ,  A ,  ( 1st `  ( f `  n
) ) ) ,  ( 2nd `  (
f `  n )
) ) )
3635, 33opeq12d 4088 . . . . . . . . . . . 12  |-  ( m  =  n  ->  <. if ( if ( ( 1st `  ( f `  m
) )  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  <_ 
( 2nd `  (
f `  m )
) ,  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) ) ,  ( 2nd `  (
f `  m )
) ) ,  ( 2nd `  ( f `
 m ) )
>.  =  <. if ( if ( ( 1st `  ( f `  n
) )  <_  A ,  A ,  ( 1st `  ( f `  n
) ) )  <_ 
( 2nd `  (
f `  n )
) ,  if ( ( 1st `  (
f `  n )
)  <_  A ,  A ,  ( 1st `  ( f `  n
) ) ) ,  ( 2nd `  (
f `  n )
) ) ,  ( 2nd `  ( f `
 n ) )
>. )
3736cbvmptv 4404 . . . . . . . . . . 11  |-  ( m  e.  NN  |->  <. if ( if ( ( 1st `  ( f `  m
) )  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  <_ 
( 2nd `  (
f `  m )
) ,  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) ) ,  ( 2nd `  (
f `  m )
) ) ,  ( 2nd `  ( f `
 m ) )
>. )  =  (
n  e.  NN  |->  <. if ( if ( ( 1st `  ( f `
 n ) )  <_  A ,  A ,  ( 1st `  (
f `  n )
) )  <_  ( 2nd `  ( f `  n ) ) ,  if ( ( 1st `  ( f `  n
) )  <_  A ,  A ,  ( 1st `  ( f `  n
) ) ) ,  ( 2nd `  (
f `  n )
) ) ,  ( 2nd `  ( f `
 n ) )
>. )
3830, 35opeq12d 4088 . . . . . . . . . . . 12  |-  ( m  =  n  ->  <. ( 1st `  ( f `  m ) ) ,  if ( if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  <_ 
( 2nd `  (
f `  m )
) ,  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) ) ,  ( 2nd `  (
f `  m )
) ) >.  =  <. ( 1st `  ( f `
 n ) ) ,  if ( if ( ( 1st `  (
f `  n )
)  <_  A ,  A ,  ( 1st `  ( f `  n
) ) )  <_ 
( 2nd `  (
f `  n )
) ,  if ( ( 1st `  (
f `  n )
)  <_  A ,  A ,  ( 1st `  ( f `  n
) ) ) ,  ( 2nd `  (
f `  n )
) ) >. )
3938cbvmptv 4404 . . . . . . . . . . 11  |-  ( m  e.  NN  |->  <. ( 1st `  ( f `  m ) ) ,  if ( if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  <_ 
( 2nd `  (
f `  m )
) ,  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) ) ,  ( 2nd `  (
f `  m )
) ) >. )  =  ( n  e.  NN  |->  <. ( 1st `  (
f `  n )
) ,  if ( if ( ( 1st `  ( f `  n
) )  <_  A ,  A ,  ( 1st `  ( f `  n
) ) )  <_ 
( 2nd `  (
f `  n )
) ,  if ( ( 1st `  (
f `  n )
)  <_  A ,  A ,  ( 1st `  ( f `  n
) ) ) ,  ( 2nd `  (
f `  n )
) ) >. )
4011, 12, 13, 14, 15, 8, 16, 17, 24, 25, 26, 27, 28, 37, 39ioombl1lem4 21064 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  /\  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( x  C_  U.
ran  ( (,)  o.  f )  /\  sup ( ran  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  f ) ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  +  y ) ) ) )  ->  ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  (
( vol* `  x )  +  y ) )
4110, 40rexlimddv 2866 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  (
( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  ( ( vol* `  x )  +  y ) )
4241ralrimiva 2820 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x
)  e.  RR ) )  ->  A. y  e.  RR+  ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  (
( vol* `  x )  +  y ) )
43 inss1 3591 . . . . . . . . . . . 12  |-  ( x  i^i  ( A (,) +oo ) )  C_  x
44 ovolsscl 20991 . . . . . . . . . . . 12  |-  ( ( ( x  i^i  ( A (,) +oo ) ) 
C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  e.  RR )
4543, 44mp3an1 1301 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( A (,) +oo ) ) )  e.  RR )
4645adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x
)  e.  RR ) )  ->  ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  e.  RR )
47 difss 3504 . . . . . . . . . . . 12  |-  ( x 
\  ( A (,) +oo ) )  C_  x
48 ovolsscl 20991 . . . . . . . . . . . 12  |-  ( ( ( x  \  ( A (,) +oo ) ) 
C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x 
\  ( A (,) +oo ) ) )  e.  RR )
4947, 48mp3an1 1301 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  ( A (,) +oo ) ) )  e.  RR )
5049adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x
)  e.  RR ) )  ->  ( vol* `  ( x  \ 
( A (,) +oo ) ) )  e.  RR )
5146, 50readdcld 9434 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x
)  e.  RR ) )  ->  ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  e.  RR )
52 simprr 756 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x
)  e.  RR ) )  ->  ( vol* `  x )  e.  RR )
53 alrple 11197 . . . . . . . . 9  |-  ( ( ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  e.  RR  /\  ( vol* `  x )  e.  RR )  -> 
( ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  ( vol* `  x )  <->  A. y  e.  RR+  (
( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  ( ( vol* `  x )  +  y ) ) )
5451, 52, 53syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x
)  e.  RR ) )  ->  ( (
( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  ( vol* `  x )  <->  A. y  e.  RR+  ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  (
( vol* `  x )  +  y ) ) )
5542, 54mpbird 232 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x
)  e.  RR ) )  ->  ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  ( vol* `  x ) )
5655expr 615 . . . . . 6  |-  ( ( A  e.  RR  /\  x  C_  RR )  -> 
( ( vol* `  x )  e.  RR  ->  ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  ( vol* `  x ) ) )
574, 56sylan2 474 . . . . 5  |-  ( ( A  e.  RR  /\  x  e.  ~P RR )  ->  ( ( vol* `  x )  e.  RR  ->  ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  ( vol* `  x ) ) )
5857ralrimiva 2820 . . . 4  |-  ( A  e.  RR  ->  A. x  e.  ~P  RR ( ( vol* `  x
)  e.  RR  ->  ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  ( vol* `  x ) ) )
59 ismbl2 21032 . . . 4  |-  ( ( A (,) +oo )  e.  dom  vol  <->  ( ( A (,) +oo )  C_  RR  /\  A. x  e. 
~P  RR ( ( vol* `  x
)  e.  RR  ->  ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  ( vol* `  x ) ) ) )
603, 58, 59sylanbrc 664 . . 3  |-  ( A  e.  RR  ->  ( A (,) +oo )  e. 
dom  vol )
61 oveq1 6119 . . . . 5  |-  ( A  = +oo  ->  ( A (,) +oo )  =  ( +oo (,) +oo ) )
62 iooid 11349 . . . . 5  |-  ( +oo (,) +oo )  =  (/)
6361, 62syl6eq 2491 . . . 4  |-  ( A  = +oo  ->  ( A (,) +oo )  =  (/) )
64 0mbl 21043 . . . 4  |-  (/)  e.  dom  vol
6563, 64syl6eqel 2531 . . 3  |-  ( A  = +oo  ->  ( A (,) +oo )  e. 
dom  vol )
66 oveq1 6119 . . . . 5  |-  ( A  = -oo  ->  ( A (,) +oo )  =  ( -oo (,) +oo ) )
67 ioomax 11391 . . . . 5  |-  ( -oo (,) +oo )  =  RR
6866, 67syl6eq 2491 . . . 4  |-  ( A  = -oo  ->  ( A (,) +oo )  =  RR )
69 rembl 21044 . . . 4  |-  RR  e.  dom  vol
7068, 69syl6eqel 2531 . . 3  |-  ( A  = -oo  ->  ( A (,) +oo )  e. 
dom  vol )
7160, 65, 703jaoi 1281 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( A (,) +oo )  e. 
dom  vol )
721, 71sylbi 195 1  |-  ( A  e.  RR*  ->  ( A (,) +oo )  e. 
dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 964    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737    \ cdif 3346    i^i cin 3348    C_ wss 3349   (/)c0 3658   ifcif 3812   ~Pcpw 3881   <.cop 3904   U.cuni 4112   class class class wbr 4313    e. cmpt 4371    X. cxp 4859   dom cdm 4861   ran crn 4862    o. ccom 4865   -->wf 5435   ` cfv 5439  (class class class)co 6112   1stc1st 6596   2ndc2nd 6597    ^m cmap 7235   supcsup 7711   RRcr 9302   1c1 9304    + caddc 9306   +oocpnf 9436   -oocmnf 9437   RR*cxr 9438    < clt 9439    <_ cle 9440    - cmin 9616   NNcn 10343   RR+crp 11012   (,)cioo 11321    seqcseq 11827   abscabs 12744   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-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  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-fal 1375  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-int 4150  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-se 4701  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-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-oi 7745  df-card 8130  df-cda 8358  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-q 10975  df-rp 11013  df-xadd 11111  df-ioo 11325  df-ico 11327  df-icc 11328  df-fz 11459  df-fzo 11570  df-fl 11663  df-seq 11828  df-exp 11887  df-hash 12125  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-rlim 12988  df-sum 13185  df-xmet 17832  df-met 17833  df-ovol 20970  df-vol 20971
This theorem is referenced by:  icombl1  21066
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