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Theorem voliunlem2 22504
Description: Lemma for voliun 22507. (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 5735 . . . . 5  |-  ( F : NN --> dom  vol  ->  ran  F  C_  dom  vol )
31, 2syl 17 . . . 4  |-  ( ph  ->  ran  F  C_  dom  vol )
4 mblss 22485 . . . . . 6  |-  ( x  e.  dom  vol  ->  x 
C_  RR )
5 selpw 3958 . . . . . 6  |-  ( x  e.  ~P RR  <->  x  C_  RR )
64, 5sylibr 216 . . . . 5  |-  ( x  e.  dom  vol  ->  x  e.  ~P RR )
76ssriv 3436 . . . 4  |-  dom  vol  C_ 
~P RR
83, 7syl6ss 3444 . . 3  |-  ( ph  ->  ran  F  C_  ~P RR )
9 sspwuni 4367 . . 3  |-  ( ran 
F  C_  ~P RR  <->  U.
ran  F  C_  RR )
108, 9sylib 200 . 2  |-  ( ph  ->  U. ran  F  C_  RR )
11 elpwi 3960 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
12 inundif 3845 . . . . . . . 8  |-  ( ( x  i^i  U. ran  F )  u.  ( x 
\  U. ran  F ) )  =  x
1312fveq2i 5868 . . . . . . 7  |-  ( vol* `  ( (
x  i^i  U. ran  F
)  u.  ( x 
\  U. ran  F ) ) )  =  ( vol* `  x
)
14 inss1 3652 . . . . . . . . 9  |-  ( x  i^i  U. ran  F
)  C_  x
15 simp2 1009 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  x  C_  RR )
1614, 15syl5ss 3443 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  i^i  U. ran  F )  C_  RR )
17 ovolsscl 22439 . . . . . . . . . 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 1351 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  U. ran  F ) )  e.  RR )
19183adant1 1026 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  e.  RR )
20 difss 3560 . . . . . . . . 9  |-  ( x 
\  U. ran  F ) 
C_  x
2120, 15syl5ss 3443 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  \  U. ran  F )  C_  RR )
22 ovolsscl 22439 . . . . . . . . . 10  |-  ( ( ( x  \  U. ran  F )  C_  x  /\  x  C_  RR  /\  ( vol* `  x
)  e.  RR )  ->  ( vol* `  ( x  \  U. ran  F ) )  e.  RR )
2320, 22mp3an1 1351 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  U. ran  F ) )  e.  RR )
24233adant1 1026 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  \  U. ran  F ) )  e.  RR )
25 ovolun 22452 . . . . . . . 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 1269 . . . . . . 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 4437 . . . . . 6  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  <_  (
( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) )
2819rexrd 9690 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  e. 
RR* )
29 nnuz 11194 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
30 1zzd 10968 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  1  e.  ZZ )
31 fveq2 5865 . . . . . . . . . . . . . . . . 17  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
3231ineq2d 3634 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  (
x  i^i  ( F `  n ) )  =  ( x  i^i  ( F `  k )
) )
3332fveq2d 5869 . . . . . . . . . . . . . . 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 5875 . . . . . . . . . . . . . . 15  |-  ( vol* `  ( x  i^i  ( F `  k
) ) )  e. 
_V
3633, 34, 35fvmpt 5948 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  ( H `  k )  =  ( vol* `  ( x  i^i  ( F `  k )
) ) )
3736adantl 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( H `  k )  =  ( vol* `  ( x  i^i  ( F `  k )
) ) )
38 inss1 3652 . . . . . . . . . . . . . . . 16  |-  ( x  i^i  ( F `  k ) )  C_  x
39 ovolsscl 22439 . . . . . . . . . . . . . . . 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 1351 . . . . . . . . . . . . . . 15  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( F `  k )
) )  e.  RR )
41403adant1 1026 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  ( F `  k )
) )  e.  RR )
4241adantr 467 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( vol* `  ( x  i^i  ( F `  k ) ) )  e.  RR )
4337, 42eqeltrd 2529 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( H `  k )  e.  RR )
4429, 30, 43serfre 12242 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  seq 1 (  +  ,  H ) : NN --> RR )
45 frn 5735 . . . . . . . . . . 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 9684 . . . . . . . . . 10  |-  RR  C_  RR*
4846, 47syl6ss 3444 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ran  seq 1
(  +  ,  H
)  C_  RR* )
49 supxrcl 11600 . . . . . . . . 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 1010 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  e.  RR )
5251, 24resubcld 10047 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  e.  RR )
5352rexrd 9690 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) )  e.  RR* )
54 iunin2 4342 . . . . . . . . . . 11  |-  U_ n  e.  NN  ( x  i^i  ( F `  n
) )  =  ( x  i^i  U_ n  e.  NN  ( F `  n ) )
55 ffn 5728 . . . . . . . . . . . . . 14  |-  ( F : NN --> dom  vol  ->  F  Fn  NN )
56 fniunfv 6152 . . . . . . . . . . . . . 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 1029 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  U_ n  e.  NN  ( F `  n )  =  U. ran  F
)
5958ineq2d 3634 . . . . . . . . . . 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 2497 . . . . . . . . . 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 5869 . . . . . . . . 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 3652 . . . . . . . . . . . 12  |-  ( x  i^i  ( F `  n ) )  C_  x
6463, 15syl5ss 3443 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( x  i^i  ( F `  n
) )  C_  RR )
6564adantr 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  n  e.  NN )  ->  (
x  i^i  ( F `  n ) )  C_  RR )
66 ovolsscl 22439 . . . . . . . . . . . . 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 1351 . . . . . . . . . . . 12  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( F `  n )
) )  e.  RR )
68673adant1 1026 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  ( F `  n )
) )  e.  RR )
6968adantr 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  n  e.  NN )  ->  ( vol* `  ( x  i^i  ( F `  n ) ) )  e.  RR )
7062, 34, 65, 69ovoliun 22458 . . . . . . . . 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 4425 . . . . . . . 8  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  ( x  i^i  U. ran  F ) )  <_  sup ( ran  seq 1
(  +  ,  H
) ,  RR* ,  <  ) )
7213ad2ant1 1029 . . . . . . . . . . . . 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 1029 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> Disj  i  e.  NN  ( F `  i )
)
7572, 74, 34, 15, 51voliunlem1 22503 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (
(  seq 1 (  +  ,  H ) `  k )  +  ( vol* `  (
x  \  U. ran  F
) ) )  <_ 
( vol* `  x ) )
7644ffvelrnda 6022 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  H ) `  k
)  e.  RR )
7724adantr 467 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( vol* `  ( x 
\  U. ran  F ) )  e.  RR )
78 simpl3 1013 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  ( vol* `  x )  e.  RR )
79 leaddsub 10090 . . . . . . . . . . . . 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 1268 . . . . . . . . . . . 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 214 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  /\  k  e.  NN )  ->  (  seq 1 (  +  ,  H ) `  k
)  <_  ( ( vol* `  x )  -  ( vol* `  ( x  \  U. ran  F ) ) ) )
8281ralrimiva 2802 . . . . . . . . . 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 5728 . . . . . . . . . . 11  |-  (  seq 1 (  +  ,  H ) : NN --> RR  ->  seq 1 (  +  ,  H )  Fn  NN )
84 breq1 4405 . . . . . . . . . . . 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 6025 . . . . . . . . . . 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 236 . . . . . . . . 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 11612 . . . . . . . . . 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 667 . . . . . . . . 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 236 . . . . . . . 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 11459 . . . . . . 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 10090 . . . . . . . 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 1268 . . . . . . 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 236 . . . . . 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 9670 . . . . . . 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 9777 . . . . . 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 933 . . . . 5  |-  ( (
ph  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  U. ran  F ) )  +  ( vol* `  (
x  \  U. ran  F
) ) ) )
98973expia 1210 . . . 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 477 . . 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 2802 . 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 22480 . 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 670 1  |-  ( ph  ->  U. ran  F  e. 
dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737    \ cdif 3401    u. cun 3402    i^i cin 3403    C_ wss 3404   ~Pcpw 3951   U.cuni 4198   U_ciun 4278  Disj wdisj 4373   class class class wbr 4402    |-> cmpt 4461   dom cdm 4834   ran crn 4835    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   supcsup 7954   RRcr 9538   1c1 9540    + caddc 9542   RR*cxr 9674    < clt 9675    <_ cle 9676    - cmin 9860   NNcn 10609    seqcseq 12213   vol*covol 22413   volcvol 22415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cc 8865  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-ovol 22416  df-vol 22418
This theorem is referenced by:  voliunlem3  22505  iunmbl  22506
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