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Theorem voliunlem2 21007
Description: Lemma for voliun 21010. (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 5560 . . . . 5  |-  ( F : NN --> dom  vol  ->  ran  F  C_  dom  vol )
31, 2syl 16 . . . 4  |-  ( ph  ->  ran  F  C_  dom  vol )
4 mblss 20989 . . . . . 6  |-  ( x  e.  dom  vol  ->  x 
C_  RR )
5 selpw 3862 . . . . . 6  |-  ( x  e.  ~P RR  <->  x  C_  RR )
64, 5sylibr 212 . . . . 5  |-  ( x  e.  dom  vol  ->  x  e.  ~P RR )
76ssriv 3355 . . . 4  |-  dom  vol  C_ 
~P RR
83, 7syl6ss 3363 . . 3  |-  ( ph  ->  ran  F  C_  ~P RR )
9 sspwuni 4251 . . 3  |-  ( ran 
F  C_  ~P RR  <->  U.
ran  F  C_  RR )
108, 9sylib 196 . 2  |-  ( ph  ->  U. ran  F  C_  RR )
11 elpwi 3864 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
12 inundif 3752 . . . . . . . 8  |-  ( ( x  i^i  U. ran  F )  u.  ( x 
\  U. ran  F ) )  =  x
1312fveq2i 5689 . . . . . . 7  |-  ( vol* `  ( (
x  i^i  U. ran  F
)  u.  ( x 
\  U. ran  F ) ) )  =  ( vol* `  x
)
14 inss1 3565 . . . . . . . . 9  |-  ( x  i^i  U. ran  F
)  C_  x
15 simp2 989 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  x  C_  RR )
1614, 15syl5ss 3362 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  i^i  U. ran  F )  C_  RR )
17 ovolsscl 20944 . . . . . . . . . 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 1301 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  U. ran  F ) )  e.  RR )
19183adant1 1006 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  e.  RR )
20 difss 3478 . . . . . . . . 9  |-  ( x 
\  U. ran  F ) 
C_  x
2120, 15syl5ss 3362 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  \  U. ran  F )  C_  RR )
22 ovolsscl 20944 . . . . . . . . . 10  |-  ( ( ( x  \  U. ran  F )  C_  x  /\  x  C_  RR  /\  ( vol* `  x
)  e.  RR )  ->  ( vol* `  ( x  \  U. ran  F ) )  e.  RR )
2320, 22mp3an1 1301 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  U. ran  F ) )  e.  RR )
24233adant1 1006 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  \  U. ran  F ) )  e.  RR )
25 ovolun 20957 . . . . . . . 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 1219 . . . . . . 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 4321 . . . . . 6  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  <_  (
( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) )
2819rexrd 9425 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  e. 
RR* )
29 nnuz 10888 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
30 1zzd 10669 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  1  e.  ZZ )
31 fveq2 5686 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
3231ineq2d 3547 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  (
x  i^i  ( F `  n ) )  =  ( x  i^i  ( F `  k )
) )
3332fveq2d 5690 . . . . . . . . . . . . . . 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 5696 . . . . . . . . . . . . . . 15  |-  ( vol* `  ( x  i^i  ( F `  k
) ) )  e. 
_V
3633, 34, 35fvmpt 5769 . . . . . . . . . . . . . 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 3565 . . . . . . . . . . . . . . . 16  |-  ( x  i^i  ( F `  k ) )  C_  x
39 ovolsscl 20944 . . . . . . . . . . . . . . . 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 1301 . . . . . . . . . . . . . . 15  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( F `  k )
) )  e.  RR )
41403adant1 1006 . . . . . . . . . . . . . 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 2512 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( H `  k )  e.  RR )
4429, 30, 43serfre 11827 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  seq 1 (  +  ,  H ) : NN --> RR )
45 frn 5560 . . . . . . . . . . 11  |-  (  seq 1 (  +  ,  H ) : NN --> RR  ->  ran  seq 1
(  +  ,  H
)  C_  RR )
4644, 45syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ran  seq 1
(  +  ,  H
)  C_  RR )
47 ressxr 9419 . . . . . . . . . 10  |-  RR  C_  RR*
4846, 47syl6ss 3363 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ran  seq 1
(  +  ,  H
)  C_  RR* )
49 supxrcl 11269 . . . . . . . . 9  |-  ( ran 
seq 1 (  +  ,  H )  C_  RR* 
->  sup ( ran  seq 1 (  +  ,  H ) ,  RR* ,  <  )  e.  RR* )
5048, 49syl 16 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  sup ( ran  seq 1 (  +  ,  H ) ,  RR* ,  <  )  e.  RR* )
51 simp3 990 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  e.  RR )
5251, 24resubcld 9768 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  e.  RR )
5352rexrd 9425 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  e.  RR* )
54 iunin2 4229 . . . . . . . . . . 11  |-  U_ n  e.  NN  ( x  i^i  ( F `  n
) )  =  ( x  i^i  U_ n  e.  NN  ( F `  n ) )
55 ffn 5554 . . . . . . . . . . . . . 14  |-  ( F : NN --> dom  vol  ->  F  Fn  NN )
56 fniunfv 5959 . . . . . . . . . . . . . 14  |-  ( F  Fn  NN  ->  U_ n  e.  NN  ( F `  n )  =  U. ran  F )
571, 55, 563syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  U_ n  e.  NN  ( F `  n )  =  U. ran  F
)
58573ad2ant1 1009 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  U_ n  e.  NN  ( F `  n )  =  U. ran  F
)
5958ineq2d 3547 . . . . . . . . . . 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 2482 . . . . . . . . . 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 5690 . . . . . . . . 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 2438 . . . . . . . . . 10  |-  seq 1
(  +  ,  H
)  =  seq 1
(  +  ,  H
)
63 inss1 3565 . . . . . . . . . . . 12  |-  ( x  i^i  ( F `  n ) )  C_  x
6463, 15syl5ss 3362 . . . . . . . . . . 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 20944 . . . . . . . . . . . . 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 1301 . . . . . . . . . . . 12  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( F `  n )
) )  e.  RR )
68673adant1 1006 . . . . . . . . . . 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 20963 . . . . . . . . 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 4309 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  <_  sup ( ran  seq 1
(  +  ,  H
) ,  RR* ,  <  ) )
7213ad2ant1 1009 . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> Disj  i  e.  NN  ( F `  i )
)
7572, 74, 34, 15, 51voliunlem1 21006 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (
(  seq 1 (  +  ,  H ) `  k )  +  ( vol* `  (
x  \  U. ran  F
) ) )  <_ 
( vol* `  x ) )
7644ffvelrnda 5838 . . . . . . . . . . . . 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 993 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( vol* `  x )  e.  RR )
79 leaddsub 9807 . . . . . . . . . . . . 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 1218 . . . . . . . . . . . 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 210 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  H ) `  k
)  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) ) )
8281ralrimiva 2794 . . . . . . . . . 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 5554 . . . . . . . . . . 11  |-  (  seq 1 (  +  ,  H ) : NN --> RR  ->  seq 1 (  +  ,  H )  Fn  NN )
84 breq1 4290 . . . . . . . . . . . 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 5841 . . . . . . . . . . 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 20 . . . . . . . . . 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 232 . . . . . . . . 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 11281 . . . . . . . . . 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 232 . . . . . . . 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 11128 . . . . . . 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 9807 . . . . . . . 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 1218 . . . . . . 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 232 . . . . . 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 9405 . . . . . . 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 9508 . . . . . 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 913 . . . . 5  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) )
98973expia 1189 . . . 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 2794 . 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 20984 . 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710    \ cdif 3320    u. cun 3321    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   U.cuni 4086   U_ciun 4166  Disj wdisj 4257   class class class wbr 4287    e. cmpt 4345   dom cdm 4835   ran crn 4836    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   supcsup 7682   RRcr 9273   1c1 9275    + caddc 9277   RR*cxr 9409    < clt 9410    <_ cle 9411    - cmin 9587   NNcn 10314    seqcseq 11798   vol*covol 20921   volcvol 20922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cc 8596  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-ovol 20923  df-vol 20924
This theorem is referenced by:  voliunlem3  21008  iunmbl  21009
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