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Theorem ioombl1 22594
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 11439 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 ioossre 11721 . . . . 5  |-  ( A (,) +oo )  C_  RR
32a1i 11 . . . 4  |-  ( A  e.  RR  ->  ( A (,) +oo )  C_  RR )
4 elpwi 3951 . . . . . 6  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
5 simplrl 778 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  x  C_  RR )
6 simplrr 779 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  ( vol* `  x )  e.  RR )
7 simpr 468 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  y  e.  RR+ )
8 eqid 2471 . . . . . . . . . . . 12  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
98ovolgelb 22511 . . . . . . . . . . 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 1292 . . . . . . . . . 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 2471 . . . . . . . . . . 11  |-  ( A (,) +oo )  =  ( A (,) +oo )
12 simplll 776 . . . . . . . . . . 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 472 . . . . . . . . . . 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 472 . . . . . . . . . . 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 770 . . . . . . . . . . 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 2471 . . . . . . . . . . 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 2471 . . . . . . . . . . 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 772 . . . . . . . . . . . 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 9648 . . . . . . . . . . . . . . 15  |-  RR  e.  _V
2019, 19xpex 6614 . . . . . . . . . . . . . 14  |-  ( RR 
X.  RR )  e. 
_V
2120inex2 4538 . . . . . . . . . . . . 13  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
22 nnex 10637 . . . . . . . . . . . . 13  |-  NN  e.  _V
2321, 22elmap 7518 . . . . . . . . . . . 12  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2418, 23sylib 201 . . . . . . . . . . 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 782 . . . . . . . . . . 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 783 . . . . . . . . . . 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 2471 . . . . . . . . . . 11  |-  ( 1st `  ( f `  n
) )  =  ( 1st `  ( f `
 n ) )
28 eqid 2471 . . . . . . . . . . 11  |-  ( 2nd `  ( f `  n
) )  =  ( 2nd `  ( f `
 n ) )
29 fveq2 5879 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  n  ->  (
f `  m )  =  ( f `  n ) )
3029fveq2d 5883 . . . . . . . . . . . . . . . . 17  |-  ( m  =  n  ->  ( 1st `  ( f `  m ) )  =  ( 1st `  (
f `  n )
) )
3130breq1d 4405 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
( 1st `  (
f `  m )
)  <_  A  <->  ( 1st `  ( f `  n
) )  <_  A
) )
3231, 30ifbieq2d 3897 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  =  if ( ( 1st `  ( f `  n
) )  <_  A ,  A ,  ( 1st `  ( f `  n
) ) ) )
3329fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  ( 2nd `  ( f `  m ) )  =  ( 2nd `  (
f `  n )
) )
3432, 33breq12d 4408 . . . . . . . . . . . . . 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 3899 . . . . . . . . . . . . 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 4166 . . . . . . . . . . . 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 4488 . . . . . . . . . . 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 4166 . . . . . . . . . . . 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 4488 . . . . . . . . . . 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 22593 . . . . . . . . . 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 2875 . . . . . . . . 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 2809 . . . . . . . 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 3643 . . . . . . . . . . . 12  |-  ( x  i^i  ( A (,) +oo ) )  C_  x
44 ovolsscl 22517 . . . . . . . . . . . 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 1377 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( A (,) +oo ) ) )  e.  RR )
4645adantl 473 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x
)  e.  RR ) )  ->  ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  e.  RR )
47 difss 3549 . . . . . . . . . . . 12  |-  ( x 
\  ( A (,) +oo ) )  C_  x
48 ovolsscl 22517 . . . . . . . . . . . 12  |-  ( ( ( x  \  ( A (,) +oo ) ) 
C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x 
\  ( A (,) +oo ) ) )  e.  RR )
4947, 48mp3an1 1377 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  ( A (,) +oo ) ) )  e.  RR )
5049adantl 473 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x
)  e.  RR ) )  ->  ( vol* `  ( x  \ 
( A (,) +oo ) ) )  e.  RR )
5146, 50readdcld 9688 . . . . . . . . 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 774 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x
)  e.  RR ) )  ->  ( vol* `  x )  e.  RR )
53 alrple 11522 . . . . . . . . 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 673 . . . . . . . 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 240 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol* `  x
)  e.  RR ) )  ->  ( ( vol* `  ( x  i^i  ( A (,) +oo ) ) )  +  ( vol* `  ( x  \  ( A (,) +oo ) ) ) )  <_  ( vol* `  x ) )
5655expr 626 . . . . . 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 482 . . . . 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 2809 . . . 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 22559 . . . 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 677 . . 3  |-  ( A  e.  RR  ->  ( A (,) +oo )  e. 
dom  vol )
61 oveq1 6315 . . . . 5  |-  ( A  = +oo  ->  ( A (,) +oo )  =  ( +oo (,) +oo ) )
62 iooid 11689 . . . . 5  |-  ( +oo (,) +oo )  =  (/)
6361, 62syl6eq 2521 . . . 4  |-  ( A  = +oo  ->  ( A (,) +oo )  =  (/) )
64 0mbl 22571 . . . 4  |-  (/)  e.  dom  vol
6563, 64syl6eqel 2557 . . 3  |-  ( A  = +oo  ->  ( A (,) +oo )  e. 
dom  vol )
66 oveq1 6315 . . . . 5  |-  ( A  = -oo  ->  ( A (,) +oo )  =  ( -oo (,) +oo ) )
67 ioomax 11734 . . . . 5  |-  ( -oo (,) +oo )  =  RR
6866, 67syl6eq 2521 . . . 4  |-  ( A  = -oo  ->  ( A (,) +oo )  =  RR )
69 rembl 22572 . . . 4  |-  RR  e.  dom  vol
7068, 69syl6eqel 2557 . . 3  |-  ( A  = -oo  ->  ( A (,) +oo )  e. 
dom  vol )
7160, 65, 703jaoi 1357 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( A (,) +oo )  e. 
dom  vol )
721, 71sylbi 200 1  |-  ( A  e.  RR*  ->  ( A (,) +oo )  e. 
dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    \/ w3o 1006    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757    \ cdif 3387    i^i cin 3389    C_ wss 3390   (/)c0 3722   ifcif 3872   ~Pcpw 3942   <.cop 3965   U.cuni 4190   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   ran crn 4840    o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811    ^m cmap 7490   supcsup 7972   RRcr 9556   1c1 9558    + caddc 9560   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   RR+crp 11325   (,)cioo 11660    seqcseq 12251   abscabs 13374   vol*covol 22491   volcvol 22493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xadd 11433  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-xmet 19040  df-met 19041  df-ovol 22494  df-vol 22496
This theorem is referenced by:  icombl1  22595
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