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Theorem voliunlem2 22583
Description: Lemma for voliun 22586. (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 5747 . . . . 5  |-  ( F : NN --> dom  vol  ->  ran  F  C_  dom  vol )
31, 2syl 17 . . . 4  |-  ( ph  ->  ran  F  C_  dom  vol )
4 mblss 22563 . . . . . 6  |-  ( x  e.  dom  vol  ->  x 
C_  RR )
5 selpw 3949 . . . . . 6  |-  ( x  e.  ~P RR  <->  x  C_  RR )
64, 5sylibr 217 . . . . 5  |-  ( x  e.  dom  vol  ->  x  e.  ~P RR )
76ssriv 3422 . . . 4  |-  dom  vol  C_ 
~P RR
83, 7syl6ss 3430 . . 3  |-  ( ph  ->  ran  F  C_  ~P RR )
9 sspwuni 4360 . . 3  |-  ( ran 
F  C_  ~P RR  <->  U.
ran  F  C_  RR )
108, 9sylib 201 . 2  |-  ( ph  ->  U. ran  F  C_  RR )
11 elpwi 3951 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
12 inundif 3836 . . . . . . . 8  |-  ( ( x  i^i  U. ran  F )  u.  ( x 
\  U. ran  F ) )  =  x
1312fveq2i 5882 . . . . . . 7  |-  ( vol* `  ( (
x  i^i  U. ran  F
)  u.  ( x 
\  U. ran  F ) ) )  =  ( vol* `  x
)
14 inss1 3643 . . . . . . . . 9  |-  ( x  i^i  U. ran  F
)  C_  x
15 simp2 1031 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  x  C_  RR )
1614, 15syl5ss 3429 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  i^i  U. ran  F )  C_  RR )
17 ovolsscl 22517 . . . . . . . . . 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 1377 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  U. ran  F ) )  e.  RR )
19183adant1 1048 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  e.  RR )
20 difss 3549 . . . . . . . . 9  |-  ( x 
\  U. ran  F ) 
C_  x
2120, 15syl5ss 3429 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  \  U. ran  F )  C_  RR )
22 ovolsscl 22517 . . . . . . . . . 10  |-  ( ( ( x  \  U. ran  F )  C_  x  /\  x  C_  RR  /\  ( vol* `  x
)  e.  RR )  ->  ( vol* `  ( x  \  U. ran  F ) )  e.  RR )
2320, 22mp3an1 1377 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  U. ran  F ) )  e.  RR )
24233adant1 1048 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  \  U. ran  F ) )  e.  RR )
25 ovolun 22530 . . . . . . . 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 1293 . . . . . . 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 4430 . . . . . 6  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  <_  (
( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) )
2819rexrd 9708 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  e. 
RR* )
29 nnuz 11218 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
30 1zzd 10992 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  1  e.  ZZ )
31 fveq2 5879 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
3231ineq2d 3625 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  (
x  i^i  ( F `  n ) )  =  ( x  i^i  ( F `  k )
) )
3332fveq2d 5883 . . . . . . . . . . . . . . 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 5889 . . . . . . . . . . . . . . 15  |-  ( vol* `  ( x  i^i  ( F `  k
) ) )  e. 
_V
3633, 34, 35fvmpt 5963 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( H `  k )  =  ( vol* `  ( x  i^i  ( F `  k )
) ) )
3736adantl 473 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( H `  k )  =  ( vol* `  ( x  i^i  ( F `  k )
) ) )
38 inss1 3643 . . . . . . . . . . . . . . . 16  |-  ( x  i^i  ( F `  k ) )  C_  x
39 ovolsscl 22517 . . . . . . . . . . . . . . . 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 1377 . . . . . . . . . . . . . . 15  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( F `  k )
) )  e.  RR )
41403adant1 1048 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  ( F `  k )
) )  e.  RR )
4241adantr 472 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( vol* `  ( x  i^i  ( F `  k ) ) )  e.  RR )
4337, 42eqeltrd 2549 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( H `  k )  e.  RR )
4429, 30, 43serfre 12280 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  seq 1 (  +  ,  H ) : NN --> RR )
45 frn 5747 . . . . . . . . . . 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 9702 . . . . . . . . . 10  |-  RR  C_  RR*
4846, 47syl6ss 3430 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ran  seq 1
(  +  ,  H
)  C_  RR* )
49 supxrcl 11625 . . . . . . . . 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 1032 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  e.  RR )
5251, 24resubcld 10068 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  e.  RR )
5352rexrd 9708 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  e.  RR* )
54 iunin2 4333 . . . . . . . . . . 11  |-  U_ n  e.  NN  ( x  i^i  ( F `  n
) )  =  ( x  i^i  U_ n  e.  NN  ( F `  n ) )
55 ffn 5739 . . . . . . . . . . . . . 14  |-  ( F : NN --> dom  vol  ->  F  Fn  NN )
56 fniunfv 6170 . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  U_ n  e.  NN  ( F `  n )  =  U. ran  F
)
5958ineq2d 3625 . . . . . . . . . . 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 2517 . . . . . . . . . 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 5883 . . . . . . . . 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 2471 . . . . . . . . . 10  |-  seq 1
(  +  ,  H
)  =  seq 1
(  +  ,  H
)
63 inss1 3643 . . . . . . . . . . . 12  |-  ( x  i^i  ( F `  n ) )  C_  x
6463, 15syl5ss 3429 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  i^i  ( F `  n
) )  C_  RR )
6564adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  n  e.  NN )  ->  (
x  i^i  ( F `  n ) )  C_  RR )
66 ovolsscl 22517 . . . . . . . . . . . . 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 1377 . . . . . . . . . . . 12  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( F `  n )
) )  e.  RR )
68673adant1 1048 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  ( F `  n )
) )  e.  RR )
6968adantr 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  n  e.  NN )  ->  ( vol* `  ( x  i^i  ( F `  n ) ) )  e.  RR )
7062, 34, 65, 69ovoliun 22536 . . . . . . . . 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 4418 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  <_  sup ( ran  seq 1
(  +  ,  H
) ,  RR* ,  <  ) )
7213ad2ant1 1051 . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> Disj  i  e.  NN  ( F `  i )
)
7572, 74, 34, 15, 51voliunlem1 22582 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (
(  seq 1 (  +  ,  H ) `  k )  +  ( vol* `  (
x  \  U. ran  F
) ) )  <_ 
( vol* `  x ) )
7644ffvelrnda 6037 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  H ) `  k
)  e.  RR )
7724adantr 472 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( vol* `  ( x 
\  U. ran  F ) )  e.  RR )
78 simpl3 1035 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( vol* `  x )  e.  RR )
79 leaddsub 10111 . . . . . . . . . . . . 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 1292 . . . . . . . . . . . 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 215 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  H ) `  k
)  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) ) )
8281ralrimiva 2809 . . . . . . . . . 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 5739 . . . . . . . . . . 11  |-  (  seq 1 (  +  ,  H ) : NN --> RR  ->  seq 1 (  +  ,  H )  Fn  NN )
84 breq1 4398 . . . . . . . . . . . 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 6040 . . . . . . . . . . 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 240 . . . . . . . . 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 11637 . . . . . . . . . 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 673 . . . . . . . . 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 240 . . . . . . . 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 11482 . . . . . . 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 10111 . . . . . . . 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 1292 . . . . . . 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 240 . . . . . 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 9688 . . . . . . 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 9794 . . . . . 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 936 . . . . 5  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) )
98973expia 1233 . . . 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 482 . . 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 2809 . 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 22558 . 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 677 1  |-  ( ph  ->  U. ran  F  e. 
dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   U.cuni 4190   U_ciun 4269  Disj wdisj 4366   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   supcsup 7972   RRcr 9556   1c1 9558    + caddc 9560   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631    seqcseq 12251   vol*covol 22491   volcvol 22493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-ovol 22494  df-vol 22496
This theorem is referenced by:  voliunlem3  22584  iunmbl  22585
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