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Theorem ioombl1 19458
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 10718 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
2 ioossre 10974 . . . . 5  |-  ( A (,)  +oo )  C_  RR
32a1i 11 . . . 4  |-  ( A  e.  RR  ->  ( A (,)  +oo )  C_  RR )
4 elpwi 3809 . . . . . 6  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
5 simplrl 738 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  x  C_  RR )
6 simplrr 739 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  ( vol * `  x )  e.  RR )
7 simpr 449 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  y  e.  RR+ )
8 eqid 2438 . . . . . . . . . . . 12  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
98ovolgelb 19378 . . . . . . . . . . 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 1185 . . . . . . . . . 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 2438 . . . . . . . . . . 11  |-  ( A (,)  +oo )  =  ( A (,)  +oo )
12 simplll 736 . . . . . . . . . . 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 453 . . . . . . . . . . 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 453 . . . . . . . . . . 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 733 . . . . . . . . . . 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 2438 . . . . . . . . . . 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 2438 . . . . . . . . . . 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 734 . . . . . . . . . . . 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 9083 . . . . . . . . . . . . . . 15  |-  RR  e.  _V
2019, 19xpex 4992 . . . . . . . . . . . . . 14  |-  ( RR 
X.  RR )  e. 
_V
2120inex2 4347 . . . . . . . . . . . . 13  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
22 nnex 10008 . . . . . . . . . . . . 13  |-  NN  e.  _V
2321, 22elmap 7044 . . . . . . . . . . . 12  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2418, 23sylib 190 . . . . . . . . . . 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 742 . . . . . . . . . . 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 743 . . . . . . . . . . 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 2438 . . . . . . . . . . 11  |-  ( 1st `  ( f `  n
) )  =  ( 1st `  ( f `
 n ) )
28 eqid 2438 . . . . . . . . . . 11  |-  ( 2nd `  ( f `  n
) )  =  ( 2nd `  ( f `
 n ) )
29 fveq2 5730 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  n  ->  (
f `  m )  =  ( f `  n ) )
3029fveq2d 5734 . . . . . . . . . . . . . . . . 17  |-  ( m  =  n  ->  ( 1st `  ( f `  m ) )  =  ( 1st `  (
f `  n )
) )
3130breq1d 4224 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
( 1st `  (
f `  m )
)  <_  A  <->  ( 1st `  ( f `  n
) )  <_  A
) )
3231, 30ifbieq2d 3761 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  =  if ( ( 1st `  ( f `  n
) )  <_  A ,  A ,  ( 1st `  ( f `  n
) ) ) )
3329fveq2d 5734 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  ( 2nd `  ( f `  m ) )  =  ( 2nd `  (
f `  n )
) )
3432, 33breq12d 4227 . . . . . . . . . . . . . 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 3763 . . . . . . . . . . . . 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 3994 . . . . . . . . . . . 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 4302 . . . . . . . . . . 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 3994 . . . . . . . . . . . 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 4302 . . . . . . . . . . 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 19457 . . . . . . . . . 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 2836 . . . . . . . . 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 2791 . . . . . . . 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 3563 . . . . . . . . . . . 12  |-  ( x  i^i  ( A (,)  +oo ) )  C_  x
44 ovolsscl 19384 . . . . . . . . . . . 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 1267 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  ( A (,)  +oo ) ) )  e.  RR )
4645adantl 454 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x
)  e.  RR ) )  ->  ( vol * `
 ( x  i^i  ( A (,)  +oo ) ) )  e.  RR )
47 difss 3476 . . . . . . . . . . . 12  |-  ( x 
\  ( A (,)  +oo ) )  C_  x
48 ovolsscl 19384 . . . . . . . . . . . 12  |-  ( ( ( x  \  ( A (,)  +oo ) )  C_  x  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  ->  ( vol * `  ( x  \  ( A (,)  +oo ) ) )  e.  RR )
4947, 48mp3an1 1267 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  ( A (,)  +oo ) ) )  e.  RR )
5049adantl 454 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x
)  e.  RR ) )  ->  ( vol * `
 ( x  \ 
( A (,)  +oo ) ) )  e.  RR )
5146, 50readdcld 9117 . . . . . . . . 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 735 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x
)  e.  RR ) )  ->  ( vol * `
 x )  e.  RR )
53 alrple 10794 . . . . . . . . 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 644 . . . . . . . 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 225 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x
)  e.  RR ) )  ->  ( ( vol * `  ( x  i^i  ( A (,)  +oo ) ) )  +  ( vol * `  ( x  \  ( A (,)  +oo ) ) ) )  <_  ( vol * `
 x ) )
5655expr 600 . . . . . 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 462 . . . . 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 2791 . . . 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 19425 . . . 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 647 . . 3  |-  ( A  e.  RR  ->  ( A (,)  +oo )  e.  dom  vol )
61 oveq1 6090 . . . . 5  |-  ( A  =  +oo  ->  ( A (,)  +oo )  =  ( 
+oo (,)  +oo ) )
62 iooid 10946 . . . . 5  |-  (  +oo (,) 
+oo )  =  (/)
6361, 62syl6eq 2486 . . . 4  |-  ( A  =  +oo  ->  ( A (,)  +oo )  =  (/) )
64 0mbl 19436 . . . 4  |-  (/)  e.  dom  vol
6563, 64syl6eqel 2526 . . 3  |-  ( A  =  +oo  ->  ( A (,)  +oo )  e.  dom  vol )
66 oveq1 6090 . . . . 5  |-  ( A  =  -oo  ->  ( A (,)  +oo )  =  ( 
-oo (,)  +oo ) )
67 ioomax 10987 . . . . 5  |-  (  -oo (,) 
+oo )  =  RR
6866, 67syl6eq 2486 . . . 4  |-  ( A  =  -oo  ->  ( A (,)  +oo )  =  RR )
69 rembl 19437 . . . 4  |-  RR  e.  dom  vol
7068, 69syl6eqel 2526 . . 3  |-  ( A  =  -oo  ->  ( A (,)  +oo )  e.  dom  vol )
7160, 65, 703jaoi 1248 . 2  |-  ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  ->  ( A (,)  +oo )  e.  dom  vol )
721, 71sylbi 189 1  |-  ( A  e.  RR*  ->  ( A (,)  +oo )  e.  dom  vol )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    \/ w3o 936    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ifcif 3741   ~Pcpw 3801   <.cop 3819   U.cuni 4017   class class class wbr 4214    e. cmpt 4268    X. cxp 4878   dom cdm 4880   ran crn 4881    o. ccom 4884   -->wf 5452   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350    ^m cmap 7020   supcsup 7447   RRcr 8991   1c1 8993    + caddc 8995    +oocpnf 9119    -oocmnf 9120   RR*cxr 9121    < clt 9122    <_ cle 9123    - cmin 9293   NNcn 10002   RR+crp 10614   (,)cioo 10918    seq cseq 11325   abscabs 12041   vol
*covol 19361   volcvol 19362
This theorem is referenced by:  icombl1  19459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xadd 10713  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285  df-sum 12482  df-xmet 16697  df-met 16698  df-ovol 19363  df-vol 19364
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