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Theorem voliunlem2 21158
Description: Lemma for voliun 21161. (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 5666 . . . . 5  |-  ( F : NN --> dom  vol  ->  ran  F  C_  dom  vol )
31, 2syl 16 . . . 4  |-  ( ph  ->  ran  F  C_  dom  vol )
4 mblss 21139 . . . . . 6  |-  ( x  e.  dom  vol  ->  x 
C_  RR )
5 selpw 3968 . . . . . 6  |-  ( x  e.  ~P RR  <->  x  C_  RR )
64, 5sylibr 212 . . . . 5  |-  ( x  e.  dom  vol  ->  x  e.  ~P RR )
76ssriv 3461 . . . 4  |-  dom  vol  C_ 
~P RR
83, 7syl6ss 3469 . . 3  |-  ( ph  ->  ran  F  C_  ~P RR )
9 sspwuni 4357 . . 3  |-  ( ran 
F  C_  ~P RR  <->  U.
ran  F  C_  RR )
108, 9sylib 196 . 2  |-  ( ph  ->  U. ran  F  C_  RR )
11 elpwi 3970 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
12 inundif 3858 . . . . . . . 8  |-  ( ( x  i^i  U. ran  F )  u.  ( x 
\  U. ran  F ) )  =  x
1312fveq2i 5795 . . . . . . 7  |-  ( vol* `  ( (
x  i^i  U. ran  F
)  u.  ( x 
\  U. ran  F ) ) )  =  ( vol* `  x
)
14 inss1 3671 . . . . . . . . 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 3468 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  i^i  U. ran  F )  C_  RR )
17 ovolsscl 21094 . . . . . . . . . 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 1302 . . . . . . . . 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 3584 . . . . . . . . 9  |-  ( x 
\  U. ran  F ) 
C_  x
2120, 15syl5ss 3468 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  \  U. ran  F )  C_  RR )
22 ovolsscl 21094 . . . . . . . . . 10  |-  ( ( ( x  \  U. ran  F )  C_  x  /\  x  C_  RR  /\  ( vol* `  x
)  e.  RR )  ->  ( vol* `  ( x  \  U. ran  F ) )  e.  RR )
2320, 22mp3an1 1302 . . . . . . . . 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 21107 . . . . . . . 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 1220 . . . . . . 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 4427 . . . . . 6  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  <_  (
( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) )
2819rexrd 9537 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  e. 
RR* )
29 nnuz 11000 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
30 1zzd 10781 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  1  e.  ZZ )
31 fveq2 5792 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
3231ineq2d 3653 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  (
x  i^i  ( F `  n ) )  =  ( x  i^i  ( F `  k )
) )
3332fveq2d 5796 . . . . . . . . . . . . . . 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 5802 . . . . . . . . . . . . . . 15  |-  ( vol* `  ( x  i^i  ( F `  k
) ) )  e. 
_V
3633, 34, 35fvmpt 5876 . . . . . . . . . . . . . 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 3671 . . . . . . . . . . . . . . . 16  |-  ( x  i^i  ( F `  k ) )  C_  x
39 ovolsscl 21094 . . . . . . . . . . . . . . . 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 1302 . . . . . . . . . . . . . . 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 2539 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( H `  k )  e.  RR )
4429, 30, 43serfre 11945 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  seq 1 (  +  ,  H ) : NN --> RR )
45 frn 5666 . . . . . . . . . . 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 9531 . . . . . . . . . 10  |-  RR  C_  RR*
4846, 47syl6ss 3469 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ran  seq 1
(  +  ,  H
)  C_  RR* )
49 supxrcl 11381 . . . . . . . . 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 9880 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  e.  RR )
5352rexrd 9537 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  e.  RR* )
54 iunin2 4335 . . . . . . . . . . 11  |-  U_ n  e.  NN  ( x  i^i  ( F `  n
) )  =  ( x  i^i  U_ n  e.  NN  ( F `  n ) )
55 ffn 5660 . . . . . . . . . . . . . 14  |-  ( F : NN --> dom  vol  ->  F  Fn  NN )
56 fniunfv 6066 . . . . . . . . . . . . . 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 3653 . . . . . . . . . . 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 2504 . . . . . . . . . 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 5796 . . . . . . . . 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 2451 . . . . . . . . . 10  |-  seq 1
(  +  ,  H
)  =  seq 1
(  +  ,  H
)
63 inss1 3671 . . . . . . . . . . . 12  |-  ( x  i^i  ( F `  n ) )  C_  x
6463, 15syl5ss 3468 . . . . . . . . . . 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 21094 . . . . . . . . . . . . 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 1302 . . . . . . . . . . . 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 21113 . . . . . . . . 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 4415 . . . . . . . 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 21157 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (
(  seq 1 (  +  ,  H ) `  k )  +  ( vol* `  (
x  \  U. ran  F
) ) )  <_ 
( vol* `  x ) )
7644ffvelrnda 5945 . . . . . . . . . . . . 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 9919 . . . . . . . . . . . . 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 1219 . . . . . . . . . . . 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 2825 . . . . . . . . . 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 5660 . . . . . . . . . . 11  |-  (  seq 1 (  +  ,  H ) : NN --> RR  ->  seq 1 (  +  ,  H )  Fn  NN )
84 breq1 4396 . . . . . . . . . . . 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 5948 . . . . . . . . . . 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 11393 . . . . . . . . . 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 11240 . . . . . . 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 9919 . . . . . . . 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 1219 . . . . . . 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 9517 . . . . . . 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 9620 . . . . . 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 1190 . . . 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 2825 . 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 21134 . 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 1370    e. wcel 1758   A.wral 2795    \ cdif 3426    u. cun 3427    i^i cin 3428    C_ wss 3429   ~Pcpw 3961   U.cuni 4192   U_ciun 4272  Disj wdisj 4363   class class class wbr 4393    |-> cmpt 4451   dom cdm 4941   ran crn 4942    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193   supcsup 7794   RRcr 9385   1c1 9387    + caddc 9389   RR*cxr 9521    < clt 9522    <_ cle 9523    - cmin 9699   NNcn 10426    seqcseq 11916   vol*covol 21071   volcvol 21072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cc 8708  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-disj 4364  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-q 11058  df-rp 11096  df-ioo 11408  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-rlim 13078  df-sum 13275  df-ovol 21073  df-vol 21074
This theorem is referenced by:  voliunlem3  21159  iunmbl  21160
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