MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ioombl1 Unicode version

Theorem ioombl1 18919
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 10458 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
2 ioossre 10712 . . . . 5  |-  ( A (,)  +oo )  C_  RR
32a1i 10 . . . 4  |-  ( A  e.  RR  ->  ( A (,)  +oo )  C_  RR )
4 elpwi 3633 . . . . . 6  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
5 simplrl 736 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  x  C_  RR )
6 simplrr 737 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  ( vol * `  x )  e.  RR )
7 simpr 447 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  /\  y  e.  RR+ )  ->  y  e.  RR+ )
8 eqid 2283 . . . . . . . . . . . 12  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
98ovolgelb 18839 . . . . . . . . . . 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 1182 . . . . . . . . . 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 2283 . . . . . . . . . . . . 13  |-  ( A (,)  +oo )  =  ( A (,)  +oo )
12 simplll 734 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 451 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 451 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 731 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 2283 . . . . . . . . . . . . 13  |-  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 2283 . . . . . . . . . . . . 13  |-  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 732 . . . . . . . . . . . . . 14  |-  ( ( ( ( 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 8828 . . . . . . . . . . . . . . . . 17  |-  RR  e.  _V
2019, 19xpex 4801 . . . . . . . . . . . . . . . 16  |-  ( RR 
X.  RR )  e. 
_V
2120inex2 4156 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
22 nnex 9752 . . . . . . . . . . . . . . 15  |-  NN  e.  _V
2321, 22elmap 6796 . . . . . . . . . . . . . 14  |-  ( f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2418, 23sylib 188 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 740 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 741 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 2283 . . . . . . . . . . . . 13  |-  ( 1st `  ( f `  n
) )  =  ( 1st `  ( f `
 n ) )
28 eqid 2283 . . . . . . . . . . . . 13  |-  ( 2nd `  ( f `  n
) )  =  ( 2nd `  ( f `
 n ) )
29 fveq2 5525 . . . . . . . . . . . . . . . . . . . 20  |-  ( m  =  n  ->  (
f `  m )  =  ( f `  n ) )
3029fveq2d 5529 . . . . . . . . . . . . . . . . . . 19  |-  ( m  =  n  ->  ( 1st `  ( f `  m ) )  =  ( 1st `  (
f `  n )
) )
3130breq1d 4033 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  n  ->  (
( 1st `  (
f `  m )
)  <_  A  <->  ( 1st `  ( f `  n
) )  <_  A
) )
3231, 30ifbieq2d 3585 . . . . . . . . . . . . . . . . 17  |-  ( m  =  n  ->  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  =  if ( ( 1st `  ( f `  n
) )  <_  A ,  A ,  ( 1st `  ( f `  n
) ) ) )
3329fveq2d 5529 . . . . . . . . . . . . . . . . 17  |-  ( m  =  n  ->  ( 2nd `  ( f `  m ) )  =  ( 2nd `  (
f `  n )
) )
3432, 33breq12d 4036 . . . . . . . . . . . . . . . 16  |-  ( 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 3587 . . . . . . . . . . . . . . 15  |-  ( 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 3804 . . . . . . . . . . . . . 14  |-  ( 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 4111 . . . . . . . . . . . . 13  |-  ( 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 3804 . . . . . . . . . . . . . 14  |-  ( 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 4111 . . . . . . . . . . . . 13  |-  ( 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 18918 . . . . . . . . . . . 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 ) ) ) )  ->  ( ( vol
* `  ( x  i^i  ( A (,)  +oo ) ) )  +  ( vol * `  ( x  \  ( A (,)  +oo ) ) ) )  <_  ( ( vol * `  x )  +  y ) )
4140expr 598 . . . . . . . . . . 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  i^i  ( A (,)  +oo ) ) )  +  ( vol * `  ( x  \  ( A (,)  +oo ) ) ) )  <_  ( ( vol * `  x )  +  y ) ) )
4241rexlimdva 2667 . . . . . . . . . 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 ) )  ->  ( ( vol
* `  ( x  i^i  ( A (,)  +oo ) ) )  +  ( vol * `  ( x  \  ( A (,)  +oo ) ) ) )  <_  ( ( vol * `  x )  +  y ) ) )
4310, 42mpd 14 . . . . . . . . 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 ) )
4443ralrimiva 2626 . . . . . . . 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 ) )
45 inss1 3389 . . . . . . . . . . . 12  |-  ( x  i^i  ( A (,)  +oo ) )  C_  x
46 ovolsscl 18845 . . . . . . . . . . . 12  |-  ( ( ( x  i^i  ( A (,)  +oo ) )  C_  x  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  ->  ( vol * `  ( x  i^i  ( A (,)  +oo ) ) )  e.  RR )
4745, 46mp3an1 1264 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  ( A (,)  +oo ) ) )  e.  RR )
4847adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x
)  e.  RR ) )  ->  ( vol * `
 ( x  i^i  ( A (,)  +oo ) ) )  e.  RR )
49 difss 3303 . . . . . . . . . . . 12  |-  ( x 
\  ( A (,)  +oo ) )  C_  x
50 ovolsscl 18845 . . . . . . . . . . . 12  |-  ( ( ( x  \  ( A (,)  +oo ) )  C_  x  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  ->  ( vol * `  ( x  \  ( A (,)  +oo ) ) )  e.  RR )
5149, 50mp3an1 1264 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  ( A (,)  +oo ) ) )  e.  RR )
5251adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x
)  e.  RR ) )  ->  ( vol * `
 ( x  \ 
( A (,)  +oo ) ) )  e.  RR )
5348, 52readdcld 8862 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x
)  e.  RR ) )  ->  ( ( vol * `  ( x  i^i  ( A (,)  +oo ) ) )  +  ( vol * `  ( x  \  ( A (,)  +oo ) ) ) )  e.  RR )
54 simprr 733 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x
)  e.  RR ) )  ->  ( vol * `
 x )  e.  RR )
55 alrple 10533 . . . . . . . . 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 ) ) )
5653, 54, 55syl2anc 642 . . . . . . . 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 ) ) )
5744, 56mpbird 223 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( x  C_  RR  /\  ( vol * `  x
)  e.  RR ) )  ->  ( ( vol * `  ( x  i^i  ( A (,)  +oo ) ) )  +  ( vol * `  ( x  \  ( A (,)  +oo ) ) ) )  <_  ( vol * `
 x ) )
5857expr 598 . . . . . 6  |-  ( ( A  e.  RR  /\  x  C_  RR )  -> 
( ( vol * `  x )  e.  RR  ->  ( ( vol * `  ( x  i^i  ( A (,)  +oo ) ) )  +  ( vol * `  ( x  \  ( A (,)  +oo ) ) ) )  <_  ( vol * `
 x ) ) )
594, 58sylan2 460 . . . . 5  |-  ( ( A  e.  RR  /\  x  e.  ~P RR )  ->  ( ( vol
* `  x )  e.  RR  ->  ( ( vol * `  ( x  i^i  ( A (,)  +oo ) ) )  +  ( vol * `  ( x  \  ( A (,)  +oo ) ) ) )  <_  ( vol * `
 x ) ) )
6059ralrimiva 2626 . . . 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 ) ) )
61 ismbl2 18886 . . . 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 ) ) ) )
623, 60, 61sylanbrc 645 . . 3  |-  ( A  e.  RR  ->  ( A (,)  +oo )  e.  dom  vol )
63 oveq1 5865 . . . . 5  |-  ( A  =  +oo  ->  ( A (,)  +oo )  =  ( 
+oo (,)  +oo ) )
64 iooid 10684 . . . . 5  |-  (  +oo (,) 
+oo )  =  (/)
6563, 64syl6eq 2331 . . . 4  |-  ( A  =  +oo  ->  ( A (,)  +oo )  =  (/) )
66 0mbl 18897 . . . 4  |-  (/)  e.  dom  vol
6765, 66syl6eqel 2371 . . 3  |-  ( A  =  +oo  ->  ( A (,)  +oo )  e.  dom  vol )
68 oveq1 5865 . . . . 5  |-  ( A  =  -oo  ->  ( A (,)  +oo )  =  ( 
-oo (,)  +oo ) )
69 ioomax 10724 . . . . 5  |-  (  -oo (,) 
+oo )  =  RR
7068, 69syl6eq 2331 . . . 4  |-  ( A  =  -oo  ->  ( A (,)  +oo )  =  RR )
71 rembl 18898 . . . 4  |-  RR  e.  dom  vol
7270, 71syl6eqel 2371 . . 3  |-  ( A  =  -oo  ->  ( A (,)  +oo )  e.  dom  vol )
7362, 67, 723jaoi 1245 . 2  |-  ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  ->  ( A (,)  +oo )  e.  dom  vol )
741, 73sylbi 187 1  |-  ( A  e.  RR*  ->  ( A (,)  +oo )  e.  dom  vol )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625   <.cop 3643   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ran crn 4690    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   supcsup 7193   RRcr 8736   1c1 8738    + caddc 8740    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   RR+crp 10354   (,)cioo 10656    seq cseq 11046   abscabs 11719   vol
*covol 18822   volcvol 18823
This theorem is referenced by:  icombl1  18920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825
  Copyright terms: Public domain W3C validator