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Theorem ioombl1 21838
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 11337 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 ioossre 11598 . . . . 5  |-  ( A (,) +oo )  C_  RR
32a1i 11 . . . 4  |-  ( A  e.  RR  ->  ( A (,) +oo )  C_  RR )
4 elpwi 4025 . . . . . 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 2467 . . . . . . . . . . . 12  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
98ovolgelb 21757 . . . . . . . . . . 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 1228 . . . . . . . . . 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 2467 . . . . . . . . . . 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 2467 . . . . . . . . . . 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 2467 . . . . . . . . . . 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 9595 . . . . . . . . . . . . . . 15  |-  RR  e.  _V
2019, 19xpex 6599 . . . . . . . . . . . . . 14  |-  ( RR 
X.  RR )  e. 
_V
2120inex2 4595 . . . . . . . . . . . . 13  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
22 nnex 10554 . . . . . . . . . . . . 13  |-  NN  e.  _V
2321, 22elmap 7459 . . . . . . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( 1st `  ( f `  n
) )  =  ( 1st `  ( f `
 n ) )
28 eqid 2467 . . . . . . . . . . 11  |-  ( 2nd `  ( f `  n
) )  =  ( 2nd `  ( f `
 n ) )
29 fveq2 5872 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  n  ->  (
f `  m )  =  ( f `  n ) )
3029fveq2d 5876 . . . . . . . . . . . . . . . . 17  |-  ( m  =  n  ->  ( 1st `  ( f `  m ) )  =  ( 1st `  (
f `  n )
) )
3130breq1d 4463 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
( 1st `  (
f `  m )
)  <_  A  <->  ( 1st `  ( f `  n
) )  <_  A
) )
3231, 30ifbieq2d 3970 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  =  if ( ( 1st `  ( f `  n
) )  <_  A ,  A ,  ( 1st `  ( f `  n
) ) ) )
3329fveq2d 5876 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  ( 2nd `  ( f `  m ) )  =  ( 2nd `  (
f `  n )
) )
3432, 33breq12d 4466 . . . . . . . . . . . . . 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 3972 . . . . . . . . . . . . 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 4227 . . . . . . . . . . . 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 4544 . . . . . . . . . . 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 4227 . . . . . . . . . . . 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 4544 . . . . . . . . . . 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 21837 . . . . . . . . . 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 2963 . . . . . . . . 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 2881 . . . . . . . 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 3723 . . . . . . . . . . . 12  |-  ( x  i^i  ( A (,) +oo ) )  C_  x
44 ovolsscl 21763 . . . . . . . . . . . 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 1311 . . . . . . . . . . 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 3636 . . . . . . . . . . . 12  |-  ( x 
\  ( A (,) +oo ) )  C_  x
48 ovolsscl 21763 . . . . . . . . . . . 12  |-  ( ( ( x  \  ( A (,) +oo ) ) 
C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x 
\  ( A (,) +oo ) ) )  e.  RR )
4947, 48mp3an1 1311 . . . . . . . . . . 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 9635 . . . . . . . . 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 11417 . . . . . . . . 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 2881 . . . 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 21804 . . . 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 6302 . . . . 5  |-  ( A  = +oo  ->  ( A (,) +oo )  =  ( +oo (,) +oo ) )
62 iooid 11569 . . . . 5  |-  ( +oo (,) +oo )  =  (/)
6361, 62syl6eq 2524 . . . 4  |-  ( A  = +oo  ->  ( A (,) +oo )  =  (/) )
64 0mbl 21816 . . . 4  |-  (/)  e.  dom  vol
6563, 64syl6eqel 2563 . . 3  |-  ( A  = +oo  ->  ( A (,) +oo )  e. 
dom  vol )
66 oveq1 6302 . . . . 5  |-  ( A  = -oo  ->  ( A (,) +oo )  =  ( -oo (,) +oo ) )
67 ioomax 11611 . . . . 5  |-  ( -oo (,) +oo )  =  RR
6866, 67syl6eq 2524 . . . 4  |-  ( A  = -oo  ->  ( A (,) +oo )  =  RR )
69 rembl 21817 . . . 4  |-  RR  e.  dom  vol
7068, 69syl6eqel 2563 . . 3  |-  ( A  = -oo  ->  ( A (,) +oo )  e. 
dom  vol )
7160, 65, 703jaoi 1291 . 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 972    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    \ cdif 3478    i^i cin 3480    C_ wss 3481   (/)c0 3790   ifcif 3945   ~Pcpw 4016   <.cop 4039   U.cuni 4251   class class class wbr 4453    |-> cmpt 4511    X. cxp 5003   dom cdm 5005   ran crn 5006    o. ccom 5009   -->wf 5590   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794    ^m cmap 7432   supcsup 7912   RRcr 9503   1c1 9505    + caddc 9507   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639    < clt 9640    <_ cle 9641    - cmin 9817   NNcn 10548   RR+crp 11232   (,)cioo 11541    seqcseq 12087   abscabs 13046   vol*covol 21740   volcvol 21741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xadd 11331  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-rlim 13291  df-sum 13488  df-xmet 18280  df-met 18281  df-ovol 21742  df-vol 21743
This theorem is referenced by:  icombl1  21839
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