MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  voliunlem2 Structured version   Unicode version

Theorem voliunlem2 22255
Description: Lemma for voliun 22258. (Contributed by Mario Carneiro, 20-Mar-2014.)
Hypotheses
Ref Expression
voliunlem.3  |-  ( ph  ->  F : NN --> dom  vol )
voliunlem.5  |-  ( ph  -> Disj  i  e.  NN  ( F `  i )
)
voliunlem.6  |-  H  =  ( n  e.  NN  |->  ( vol* `  (
x  i^i  ( F `  n ) ) ) )
Assertion
Ref Expression
voliunlem2  |-  ( ph  ->  U. ran  F  e. 
dom  vol )
Distinct variable groups:    i, n, x, F    ph, n, x
Allowed substitution hints:    ph( i)    H( x, i, n)

Proof of Theorem voliunlem2
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 voliunlem.3 . . . . 5  |-  ( ph  ->  F : NN --> dom  vol )
2 frn 5722 . . . . 5  |-  ( F : NN --> dom  vol  ->  ran  F  C_  dom  vol )
31, 2syl 17 . . . 4  |-  ( ph  ->  ran  F  C_  dom  vol )
4 mblss 22236 . . . . . 6  |-  ( x  e.  dom  vol  ->  x 
C_  RR )
5 selpw 3964 . . . . . 6  |-  ( x  e.  ~P RR  <->  x  C_  RR )
64, 5sylibr 214 . . . . 5  |-  ( x  e.  dom  vol  ->  x  e.  ~P RR )
76ssriv 3448 . . . 4  |-  dom  vol  C_ 
~P RR
83, 7syl6ss 3456 . . 3  |-  ( ph  ->  ran  F  C_  ~P RR )
9 sspwuni 4362 . . 3  |-  ( ran 
F  C_  ~P RR  <->  U.
ran  F  C_  RR )
108, 9sylib 198 . 2  |-  ( ph  ->  U. ran  F  C_  RR )
11 elpwi 3966 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
12 inundif 3852 . . . . . . . 8  |-  ( ( x  i^i  U. ran  F )  u.  ( x 
\  U. ran  F ) )  =  x
1312fveq2i 5854 . . . . . . 7  |-  ( vol* `  ( (
x  i^i  U. ran  F
)  u.  ( x 
\  U. ran  F ) ) )  =  ( vol* `  x
)
14 inss1 3661 . . . . . . . . 9  |-  ( x  i^i  U. ran  F
)  C_  x
15 simp2 1000 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  x  C_  RR )
1614, 15syl5ss 3455 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  i^i  U. ran  F )  C_  RR )
17 ovolsscl 22191 . . . . . . . . . 10  |-  ( ( ( x  i^i  U. ran  F )  C_  x  /\  x  C_  RR  /\  ( vol* `  x
)  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  e.  RR )
1814, 17mp3an1 1315 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  U. ran  F ) )  e.  RR )
19183adant1 1017 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  e.  RR )
20 difss 3572 . . . . . . . . 9  |-  ( x 
\  U. ran  F ) 
C_  x
2120, 15syl5ss 3455 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  \  U. ran  F )  C_  RR )
22 ovolsscl 22191 . . . . . . . . . 10  |-  ( ( ( x  \  U. ran  F )  C_  x  /\  x  C_  RR  /\  ( vol* `  x
)  e.  RR )  ->  ( vol* `  ( x  \  U. ran  F ) )  e.  RR )
2320, 22mp3an1 1315 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  U. ran  F ) )  e.  RR )
24233adant1 1017 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  \  U. ran  F ) )  e.  RR )
25 ovolun 22204 . . . . . . . 8  |-  ( ( ( ( x  i^i  U. ran  F )  C_  RR  /\  ( vol* `  ( x  i^i  U. ran  F ) )  e.  RR )  /\  (
( x  \  U. ran  F )  C_  RR  /\  ( vol* `  ( x  \  U. ran  F ) )  e.  RR ) )  ->  ( vol* `  ( ( x  i^i  U. ran  F )  u.  ( x 
\  U. ran  F ) ) )  <_  (
( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) )
2616, 19, 21, 24, 25syl22anc 1233 . . . . . . 7  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( ( x  i^i  U. ran  F )  u.  ( x  \  U. ran  F ) ) )  <_  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  ( x  \ 
U. ran  F )
) ) )
2713, 26syl5eqbrr 4431 . . . . . 6  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  <_  (
( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) )
2819rexrd 9675 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  e. 
RR* )
29 nnuz 11164 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
30 1zzd 10938 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  1  e.  ZZ )
31 fveq2 5851 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
3231ineq2d 3643 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  (
x  i^i  ( F `  n ) )  =  ( x  i^i  ( F `  k )
) )
3332fveq2d 5855 . . . . . . . . . . . . . . 15  |-  ( n  =  k  ->  ( vol* `  ( x  i^i  ( F `  n ) ) )  =  ( vol* `  ( x  i^i  ( F `  k )
) ) )
34 voliunlem.6 . . . . . . . . . . . . . . 15  |-  H  =  ( n  e.  NN  |->  ( vol* `  (
x  i^i  ( F `  n ) ) ) )
35 fvex 5861 . . . . . . . . . . . . . . 15  |-  ( vol* `  ( x  i^i  ( F `  k
) ) )  e. 
_V
3633, 34, 35fvmpt 5934 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( H `  k )  =  ( vol* `  ( x  i^i  ( F `  k )
) ) )
3736adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( H `  k )  =  ( vol* `  ( x  i^i  ( F `  k )
) ) )
38 inss1 3661 . . . . . . . . . . . . . . . 16  |-  ( x  i^i  ( F `  k ) )  C_  x
39 ovolsscl 22191 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  i^i  ( F `  k )
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( F `  k )
) )  e.  RR )
4038, 39mp3an1 1315 . . . . . . . . . . . . . . 15  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( F `  k )
) )  e.  RR )
41403adant1 1017 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  ( F `  k )
) )  e.  RR )
4241adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( vol* `  ( x  i^i  ( F `  k ) ) )  e.  RR )
4337, 42eqeltrd 2492 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( H `  k )  e.  RR )
4429, 30, 43serfre 12182 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  seq 1 (  +  ,  H ) : NN --> RR )
45 frn 5722 . . . . . . . . . . 11  |-  (  seq 1 (  +  ,  H ) : NN --> RR  ->  ran  seq 1
(  +  ,  H
)  C_  RR )
4644, 45syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ran  seq 1
(  +  ,  H
)  C_  RR )
47 ressxr 9669 . . . . . . . . . 10  |-  RR  C_  RR*
4846, 47syl6ss 3456 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ran  seq 1
(  +  ,  H
)  C_  RR* )
49 supxrcl 11561 . . . . . . . . 9  |-  ( ran 
seq 1 (  +  ,  H )  C_  RR* 
->  sup ( ran  seq 1 (  +  ,  H ) ,  RR* ,  <  )  e.  RR* )
5048, 49syl 17 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  sup ( ran  seq 1 (  +  ,  H ) ,  RR* ,  <  )  e.  RR* )
51 simp3 1001 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  e.  RR )
5251, 24resubcld 10030 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  e.  RR )
5352rexrd 9675 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  e.  RR* )
54 iunin2 4337 . . . . . . . . . . 11  |-  U_ n  e.  NN  ( x  i^i  ( F `  n
) )  =  ( x  i^i  U_ n  e.  NN  ( F `  n ) )
55 ffn 5716 . . . . . . . . . . . . . 14  |-  ( F : NN --> dom  vol  ->  F  Fn  NN )
56 fniunfv 6142 . . . . . . . . . . . . . 14  |-  ( F  Fn  NN  ->  U_ n  e.  NN  ( F `  n )  =  U. ran  F )
571, 55, 563syl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  U_ n  e.  NN  ( F `  n )  =  U. ran  F
)
58573ad2ant1 1020 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  U_ n  e.  NN  ( F `  n )  =  U. ran  F
)
5958ineq2d 3643 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  i^i  U_ n  e.  NN  ( F `  n ) )  =  ( x  i^i  U. ran  F
) )
6054, 59syl5eq 2457 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  U_ n  e.  NN  ( x  i^i  ( F `  n )
)  =  ( x  i^i  U. ran  F
) )
6160fveq2d 5855 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  U_ n  e.  NN  ( x  i^i  ( F `  n )
) )  =  ( vol* `  (
x  i^i  U. ran  F
) ) )
62 eqid 2404 . . . . . . . . . 10  |-  seq 1
(  +  ,  H
)  =  seq 1
(  +  ,  H
)
63 inss1 3661 . . . . . . . . . . . 12  |-  ( x  i^i  ( F `  n ) )  C_  x
6463, 15syl5ss 3455 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  i^i  ( F `  n
) )  C_  RR )
6564adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  n  e.  NN )  ->  (
x  i^i  ( F `  n ) )  C_  RR )
66 ovolsscl 22191 . . . . . . . . . . . . 13  |-  ( ( ( x  i^i  ( F `  n )
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( F `  n )
) )  e.  RR )
6763, 66mp3an1 1315 . . . . . . . . . . . 12  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( F `  n )
) )  e.  RR )
68673adant1 1017 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  ( F `  n )
) )  e.  RR )
6968adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  n  e.  NN )  ->  ( vol* `  ( x  i^i  ( F `  n ) ) )  e.  RR )
7062, 34, 65, 69ovoliun 22210 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  U_ n  e.  NN  ( x  i^i  ( F `  n )
) )  <_  sup ( ran  seq 1 (  +  ,  H ) ,  RR* ,  <  )
)
7161, 70eqbrtrrd 4419 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  <_  sup ( ran  seq 1
(  +  ,  H
) ,  RR* ,  <  ) )
7213ad2ant1 1020 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  F : NN --> dom  vol )
73 voliunlem.5 . . . . . . . . . . . . . 14  |-  ( ph  -> Disj  i  e.  NN  ( F `  i )
)
74733ad2ant1 1020 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> Disj  i  e.  NN  ( F `  i )
)
7572, 74, 34, 15, 51voliunlem1 22254 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (
(  seq 1 (  +  ,  H ) `  k )  +  ( vol* `  (
x  \  U. ran  F
) ) )  <_ 
( vol* `  x ) )
7644ffvelrnda 6011 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  H ) `  k
)  e.  RR )
7724adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( vol* `  ( x 
\  U. ran  F ) )  e.  RR )
78 simpl3 1004 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( vol* `  x )  e.  RR )
79 leaddsub 10071 . . . . . . . . . . . . 13  |-  ( ( (  seq 1 (  +  ,  H ) `
 k )  e.  RR  /\  ( vol* `  ( x  \ 
U. ran  F )
)  e.  RR  /\  ( vol* `  x
)  e.  RR )  ->  ( ( (  seq 1 (  +  ,  H ) `  k )  +  ( vol* `  (
x  \  U. ran  F
) ) )  <_ 
( vol* `  x )  <->  (  seq 1 (  +  ,  H ) `  k
)  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) ) ) )
8076, 77, 78, 79syl3anc 1232 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (
( (  seq 1
(  +  ,  H
) `  k )  +  ( vol* `  ( x  \  U. ran  F ) ) )  <_  ( vol* `  x )  <->  (  seq 1 (  +  ,  H ) `  k
)  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) ) ) )
8175, 80mpbid 212 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  H ) `  k
)  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) ) )
8281ralrimiva 2820 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  A. k  e.  NN  (  seq 1 (  +  ,  H ) `  k )  <_  (
( vol* `  x )  -  ( vol* `  ( x 
\  U. ran  F ) ) ) )
83 ffn 5716 . . . . . . . . . . 11  |-  (  seq 1 (  +  ,  H ) : NN --> RR  ->  seq 1 (  +  ,  H )  Fn  NN )
84 breq1 4400 . . . . . . . . . . . 12  |-  ( z  =  (  seq 1
(  +  ,  H
) `  k )  ->  ( z  <_  (
( vol* `  x )  -  ( vol* `  ( x 
\  U. ran  F ) ) )  <->  (  seq 1 (  +  ,  H ) `  k
)  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) ) ) )
8584ralrn 6014 . . . . . . . . . . 11  |-  (  seq 1 (  +  ,  H )  Fn  NN  ->  ( A. z  e. 
ran  seq 1 (  +  ,  H ) z  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  <->  A. k  e.  NN  (  seq 1 (  +  ,  H ) `  k )  <_  (
( vol* `  x )  -  ( vol* `  ( x 
\  U. ran  F ) ) ) ) )
8644, 83, 853syl 18 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( A. z  e.  ran  seq 1 (  +  ,  H ) z  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  <->  A. k  e.  NN  (  seq 1 (  +  ,  H ) `  k )  <_  (
( vol* `  x )  -  ( vol* `  ( x 
\  U. ran  F ) ) ) ) )
8782, 86mpbird 234 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  A. z  e.  ran  seq 1 (  +  ,  H ) z  <_ 
( ( vol* `  x )  -  ( vol* `  ( x 
\  U. ran  F ) ) ) )
88 supxrleub 11573 . . . . . . . . . 10  |-  ( ( ran  seq 1 (  +  ,  H ) 
C_  RR*  /\  ( ( vol* `  x
)  -  ( vol* `  ( x  \ 
U. ran  F )
) )  e.  RR* )  ->  ( sup ( ran  seq 1 (  +  ,  H ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  -  ( vol* `  ( x 
\  U. ran  F ) ) )  <->  A. z  e.  ran  seq 1 (  +  ,  H ) z  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) ) ) )
8948, 53, 88syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( sup ( ran  seq 1 (  +  ,  H ) , 
RR* ,  <  )  <_ 
( ( vol* `  x )  -  ( vol* `  ( x 
\  U. ran  F ) ) )  <->  A. z  e.  ran  seq 1 (  +  ,  H ) z  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) ) ) )
9087, 89mpbird 234 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  sup ( ran  seq 1 (  +  ,  H ) ,  RR* ,  <  )  <_  (
( vol* `  x )  -  ( vol* `  ( x 
\  U. ran  F ) ) ) )
9128, 50, 53, 71, 90xrletrd 11420 . . . . . . 7  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  <_ 
( ( vol* `  x )  -  ( vol* `  ( x 
\  U. ran  F ) ) ) )
92 leaddsub 10071 . . . . . . . 8  |-  ( ( ( vol* `  ( x  i^i  U. ran  F ) )  e.  RR  /\  ( vol* `  ( x  \  U. ran  F ) )  e.  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( ( vol* `  (
x  i^i  U. ran  F
) )  +  ( vol* `  (
x  \  U. ran  F
) ) )  <_ 
( vol* `  x )  <->  ( vol* `  ( x  i^i  U. ran  F ) )  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) ) ) )
9319, 24, 51, 92syl3anc 1232 . . . . . . 7  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( ( vol* `  (
x  i^i  U. ran  F
) )  +  ( vol* `  (
x  \  U. ran  F
) ) )  <_ 
( vol* `  x )  <->  ( vol* `  ( x  i^i  U. ran  F ) )  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) ) ) )
9491, 93mpbird 234 . . . . . 6  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  ( x  \ 
U. ran  F )
) )  <_  ( vol* `  x ) )
9519, 24readdcld 9655 . . . . . . 7  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  ( x  \ 
U. ran  F )
) )  e.  RR )
9651, 95letri3d 9761 . . . . . 6  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  x )  =  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  ( x  \ 
U. ran  F )
) )  <->  ( ( vol* `  x )  <_  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  ( x  \ 
U. ran  F )
) )  /\  (
( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) )  <_ 
( vol* `  x ) ) ) )
9727, 94, 96mpbir2and 925 . . . . 5  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) )
98973expia 1201 . . . 4  |-  ( (
ph  /\  x  C_  RR )  ->  ( ( vol* `  x )  e.  RR  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  ( x  \  U. ran  F ) ) ) ) )
9911, 98sylan2 474 . . 3  |-  ( (
ph  /\  x  e.  ~P RR )  ->  (
( vol* `  x )  e.  RR  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) ) )
10099ralrimiva 2820 . 2  |-  ( ph  ->  A. x  e.  ~P  RR ( ( vol* `  x )  e.  RR  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) ) )
101 ismbl 22231 . 2  |-  ( U. ran  F  e.  dom  vol  <->  ( U. ran  F  C_  RR  /\ 
A. x  e.  ~P  RR ( ( vol* `  x )  e.  RR  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) ) ) )
10210, 100, 101sylanbrc 664 1  |-  ( ph  ->  U. ran  F  e. 
dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   A.wral 2756    \ cdif 3413    u. cun 3414    i^i cin 3415    C_ wss 3416   ~Pcpw 3957   U.cuni 4193   U_ciun 4273  Disj wdisj 4368   class class class wbr 4397    |-> cmpt 4455   dom cdm 4825   ran crn 4826    Fn wfn 5566   -->wf 5567   ` cfv 5571  (class class class)co 6280   supcsup 7936   RRcr 9523   1c1 9525    + caddc 9527   RR*cxr 9659    < clt 9660    <_ cle 9661    - cmin 9843   NNcn 10578    seqcseq 12153   vol*covol 22168   volcvol 22169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cc 8849  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-disj 4369  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-oi 7971  df-card 8354  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-q 11230  df-rp 11268  df-ioo 11588  df-ico 11590  df-icc 11591  df-fz 11729  df-fzo 11857  df-fl 11968  df-seq 12154  df-exp 12213  df-hash 12455  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-clim 13462  df-rlim 13463  df-sum 13660  df-ovol 22170  df-vol 22171
This theorem is referenced by:  voliunlem3  22256  iunmbl  22257
  Copyright terms: Public domain W3C validator