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

Theorem ioombl1 18935
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 10474 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
2 ioossre 10728 . . . . 5  |-  ( A (,)  +oo )  C_  RR
32a1i 10 . . . 4  |-  ( A  e.  RR  ->  ( A (,)  +oo )  C_  RR )
4 elpwi 3646 . . . . . 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 2296 . . . . . . . . . . . 12  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
98ovolgelb 18855 . . . . . . . . . . 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 2296 . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . 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 8844 . . . . . . . . . . . . . . . . 17  |-  RR  e.  _V
2019, 19xpex 4817 . . . . . . . . . . . . . . . 16  |-  ( RR 
X.  RR )  e. 
_V
2120inex2 4172 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
22 nnex 9768 . . . . . . . . . . . . . . 15  |-  NN  e.  _V
2321, 22elmap 6812 . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . 13  |-  ( 1st `  ( f `  n
) )  =  ( 1st `  ( f `
 n ) )
28 eqid 2296 . . . . . . . . . . . . 13  |-  ( 2nd `  ( f `  n
) )  =  ( 2nd `  ( f `
 n ) )
29 fveq2 5541 . . . . . . . . . . . . . . . . . . . 20  |-  ( m  =  n  ->  (
f `  m )  =  ( f `  n ) )
3029fveq2d 5545 . . . . . . . . . . . . . . . . . . 19  |-  ( m  =  n  ->  ( 1st `  ( f `  m ) )  =  ( 1st `  (
f `  n )
) )
3130breq1d 4049 . . . . . . . . . . . . . . . . . 18  |-  ( m  =  n  ->  (
( 1st `  (
f `  m )
)  <_  A  <->  ( 1st `  ( f `  n
) )  <_  A
) )
3231, 30ifbieq2d 3598 . . . . . . . . . . . . . . . . 17  |-  ( m  =  n  ->  if ( ( 1st `  (
f `  m )
)  <_  A ,  A ,  ( 1st `  ( f `  m
) ) )  =  if ( ( 1st `  ( f `  n
) )  <_  A ,  A ,  ( 1st `  ( f `  n
) ) ) )
3329fveq2d 5545 . . . . . . . . . . . . . . . . 17  |-  ( m  =  n  ->  ( 2nd `  ( f `  m ) )  =  ( 2nd `  (
f `  n )
) )
3432, 33breq12d 4052 . . . . . . . . . . . . . . . 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 3600 . . . . . . . . . . . . . . 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 3820 . . . . . . . . . . . . . 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 4127 . . . . . . . . . . . . 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 3820 . . . . . . . . . . . . . 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 4127 . . . . . . . . . . . . 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 18934 . . . . . . . . . . . 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 2680 . . . . . . . . . 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 2639 . . . . . . . 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 3402 . . . . . . . . . . . 12  |-  ( x  i^i  ( A (,)  +oo ) )  C_  x
46 ovolsscl 18861 . . . . . . . . . . . 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 3316 . . . . . . . . . . . 12  |-  ( x 
\  ( A (,)  +oo ) )  C_  x
50 ovolsscl 18861 . . . . . . . . . . . 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 8878 . . . . . . . . 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 10549 . . . . . . . . 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 2639 . . . 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 18902 . . . 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 5881 . . . . 5  |-  ( A  =  +oo  ->  ( A (,)  +oo )  =  ( 
+oo (,)  +oo ) )
64 iooid 10700 . . . . 5  |-  (  +oo (,) 
+oo )  =  (/)
6563, 64syl6eq 2344 . . . 4  |-  ( A  =  +oo  ->  ( A (,)  +oo )  =  (/) )
66 0mbl 18913 . . . 4  |-  (/)  e.  dom  vol
6765, 66syl6eqel 2384 . . 3  |-  ( A  =  +oo  ->  ( A (,)  +oo )  e.  dom  vol )
68 oveq1 5881 . . . . 5  |-  ( A  =  -oo  ->  ( A (,)  +oo )  =  ( 
-oo (,)  +oo ) )
69 ioomax 10740 . . . . 5  |-  (  -oo (,) 
+oo )  =  RR
7068, 69syl6eq 2344 . . . 4  |-  ( A  =  -oo  ->  ( A (,)  +oo )  =  RR )
71 rembl 18914 . . . 4  |-  RR  e.  dom  vol
7270, 71syl6eqel 2384 . . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638   <.cop 3656   U.cuni 3843   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   dom cdm 4705   ran crn 4706    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    ^m cmap 6788   supcsup 7209   RRcr 8752   1c1 8754    + caddc 8756    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   RR+crp 10370   (,)cioo 10672    seq cseq 11062   abscabs 11735   vol
*covol 18838   volcvol 18839
This theorem is referenced by:  icombl1  18936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xadd 10469  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-xmet 16389  df-met 16390  df-ovol 18840  df-vol 18841
  Copyright terms: Public domain W3C validator