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Theorem ovoliunlem1 22203
Description: Lemma for ovoliun 22206. (Contributed by Mario Carneiro, 12-Jun-2014.)
Hypotheses
Ref Expression
ovoliun.t  |-  T  =  seq 1 (  +  ,  G )
ovoliun.g  |-  G  =  ( n  e.  NN  |->  ( vol* `  A
) )
ovoliun.a  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
ovoliun.v  |-  ( (
ph  /\  n  e.  NN )  ->  ( vol* `  A )  e.  RR )
ovoliun.r  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
ovoliun.b  |-  ( ph  ->  B  e.  RR+ )
ovoliun.s  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  ( F `  n ) ) )
ovoliun.u  |-  U  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  H ) )
ovoliun.h  |-  H  =  ( k  e.  NN  |->  ( ( F `  ( 1st `  ( J `
 k ) ) ) `  ( 2nd `  ( J `  k
) ) ) )
ovoliun.j  |-  ( ph  ->  J : NN -1-1-onto-> ( NN  X.  NN ) )
ovoliun.f  |-  ( ph  ->  F : NN --> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
ovoliun.x1  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  U.
ran  ( (,)  o.  ( F `  n ) ) )
ovoliun.x2  |-  ( (
ph  /\  n  e.  NN )  ->  sup ( ran  S ,  RR* ,  <  )  <_  ( ( vol* `  A )  +  ( B  / 
( 2 ^ n
) ) ) )
ovoliun.k  |-  ( ph  ->  K  e.  NN )
ovoliun.l1  |-  ( ph  ->  L  e.  ZZ )
ovoliun.l2  |-  ( ph  ->  A. w  e.  ( 1 ... K ) ( 1st `  ( J `  w )
)  <_  L )
Assertion
Ref Expression
ovoliunlem1  |-  ( ph  ->  ( U `  K
)  <_  ( sup ( ran  T ,  RR* ,  <  )  +  B
) )
Distinct variable groups:    A, k    k, n, B    k, F, n    w, k, J, n   
n, K, w    k, L, n, w    n, H    ph, k, n    S, k   
k, G    T, k    n, G    T, n
Allowed substitution hints:    ph( w)    A( w, n)    B( w)    S( w, n)    T( w)    U( w, k, n)    F( w)    G( w)    H( w, k)    K( k)

Proof of Theorem ovoliunlem1
Dummy variables  j  m  x  y  z 
i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5848 . . . . . . . . 9  |-  ( j  =  ( J `  m )  ->  ( 1st `  j )  =  ( 1st `  ( J `  m )
) )
21fveq2d 5852 . . . . . . . 8  |-  ( j  =  ( J `  m )  ->  ( F `  ( 1st `  j ) )  =  ( F `  ( 1st `  ( J `  m ) ) ) )
3 fveq2 5848 . . . . . . . 8  |-  ( j  =  ( J `  m )  ->  ( 2nd `  j )  =  ( 2nd `  ( J `  m )
) )
42, 3fveq12d 5854 . . . . . . 7  |-  ( j  =  ( J `  m )  ->  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) )  =  ( ( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )
54fveq2d 5852 . . . . . 6  |-  ( j  =  ( J `  m )  ->  ( 2nd `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  =  ( 2nd `  ( ( F `  ( 1st `  ( J `
 m ) ) ) `  ( 2nd `  ( J `  m
) ) ) ) )
64fveq2d 5852 . . . . . 6  |-  ( j  =  ( J `  m )  ->  ( 1st `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  =  ( 1st `  ( ( F `  ( 1st `  ( J `
 m ) ) ) `  ( 2nd `  ( J `  m
) ) ) ) )
75, 6oveq12d 6295 . . . . 5  |-  ( j  =  ( J `  m )  ->  (
( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  =  ( ( 2nd `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )  -  ( 1st `  ( ( F `  ( 1st `  ( J `  m
) ) ) `  ( 2nd `  ( J `
 m ) ) ) ) ) )
8 fzfid 12122 . . . . 5  |-  ( ph  ->  ( 1 ... K
)  e.  Fin )
9 ovoliun.j . . . . . . 7  |-  ( ph  ->  J : NN -1-1-onto-> ( NN  X.  NN ) )
10 f1of1 5797 . . . . . . 7  |-  ( J : NN -1-1-onto-> ( NN  X.  NN )  ->  J : NN -1-1-> ( NN  X.  NN ) )
119, 10syl 17 . . . . . 6  |-  ( ph  ->  J : NN -1-1-> ( NN  X.  NN ) )
12 elfznn 11766 . . . . . . 7  |-  ( m  e.  ( 1 ... K )  ->  m  e.  NN )
1312ssriv 3445 . . . . . 6  |-  ( 1 ... K )  C_  NN
14 f1ores 5812 . . . . . 6  |-  ( ( J : NN -1-1-> ( NN  X.  NN )  /\  ( 1 ... K )  C_  NN )  ->  ( J  |`  ( 1 ... K
) ) : ( 1 ... K ) -1-1-onto-> ( J " ( 1 ... K ) ) )
1511, 13, 14sylancl 660 . . . . 5  |-  ( ph  ->  ( J  |`  (
1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " (
1 ... K ) ) )
16 fvres 5862 . . . . . 6  |-  ( m  e.  ( 1 ... K )  ->  (
( J  |`  (
1 ... K ) ) `
 m )  =  ( J `  m
) )
1716adantl 464 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( J  |`  (
1 ... K ) ) `
 m )  =  ( J `  m
) )
18 inss2 3659 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
19 ovoliun.f . . . . . . . . . . . . 13  |-  ( ph  ->  F : NN --> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
2019adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  F : NN
--> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
21 imassrn 5167 . . . . . . . . . . . . . . 15  |-  ( J
" ( 1 ... K ) )  C_  ran  J
22 f1of 5798 . . . . . . . . . . . . . . . . 17  |-  ( J : NN -1-1-onto-> ( NN  X.  NN )  ->  J : NN --> ( NN  X.  NN ) )
239, 22syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  J : NN --> ( NN 
X.  NN ) )
24 frn 5719 . . . . . . . . . . . . . . . 16  |-  ( J : NN --> ( NN 
X.  NN )  ->  ran  J  C_  ( NN  X.  NN ) )
2523, 24syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  J  C_  ( NN  X.  NN ) )
2621, 25syl5ss 3452 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  ( NN  X.  NN ) )
2726sselda 3441 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  j  e.  ( NN  X.  NN ) )
28 xp1st 6813 . . . . . . . . . . . . 13  |-  ( j  e.  ( NN  X.  NN )  ->  ( 1st `  j )  e.  NN )
2927, 28syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  e.  NN )
3020, 29ffvelrnd 6009 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( F `  ( 1st `  j
) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
31 reex 9612 . . . . . . . . . . . . . 14  |-  RR  e.  _V
3231, 31xpex 6585 . . . . . . . . . . . . 13  |-  ( RR 
X.  RR )  e. 
_V
3332inex2 4535 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
34 nnex 10581 . . . . . . . . . . . 12  |-  NN  e.  _V
3533, 34elmap 7484 . . . . . . . . . . 11  |-  ( ( F `  ( 1st `  j ) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  ( F `  ( 1st `  j ) ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
3630, 35sylib 196 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( F `  ( 1st `  j
) ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
37 xp2nd 6814 . . . . . . . . . . 11  |-  ( j  e.  ( NN  X.  NN )  ->  ( 2nd `  j )  e.  NN )
3827, 37syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 2nd `  j )  e.  NN )
3936, 38ffvelrnd 6009 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
4018, 39sseldi 3439 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) )  e.  ( RR 
X.  RR ) )
41 xp2nd 6814 . . . . . . . 8  |-  ( ( ( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) )  e.  ( RR  X.  RR )  ->  ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  e.  RR )
4240, 41syl 17 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 2nd `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) )  e.  RR )
43 xp1st 6813 . . . . . . . 8  |-  ( ( ( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) )  e.  ( RR  X.  RR )  ->  ( 1st `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  e.  RR )
4440, 43syl 17 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) )  e.  RR )
4542, 44resubcld 10027 . . . . . 6  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( 2nd `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  e.  RR )
4645recnd 9651 . . . . 5  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( 2nd `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  e.  CC )
477, 8, 15, 17, 46fsumf1o 13692 . . . 4  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  = 
sum_ m  e.  (
1 ... K ) ( ( 2nd `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )  -  ( 1st `  ( ( F `  ( 1st `  ( J `  m
) ) ) `  ( 2nd `  ( J `
 m ) ) ) ) ) )
4819adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  F : NN
--> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
4923ffvelrnda 6008 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  ( J `
 k )  e.  ( NN  X.  NN ) )
50 xp1st 6813 . . . . . . . . . . . 12  |-  ( ( J `  k )  e.  ( NN  X.  NN )  ->  ( 1st `  ( J `  k
) )  e.  NN )
5149, 50syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1st `  ( J `  k
) )  e.  NN )
5248, 51ffvelrnd 6009 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 ( 1st `  ( J `  k )
) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
5333, 34elmap 7484 . . . . . . . . . 10  |-  ( ( F `  ( 1st `  ( J `  k
) ) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  ( F `  ( 1st `  ( J `  k ) ) ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
5452, 53sylib 196 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 ( 1st `  ( J `  k )
) ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
55 xp2nd 6814 . . . . . . . . . 10  |-  ( ( J `  k )  e.  ( NN  X.  NN )  ->  ( 2nd `  ( J `  k
) )  e.  NN )
5649, 55syl 17 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( 2nd `  ( J `  k
) )  e.  NN )
5754, 56ffvelrnd 6009 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  ( 1st `  ( J `  k
) ) ) `  ( 2nd `  ( J `
 k ) ) )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
58 ovoliun.h . . . . . . . 8  |-  H  =  ( k  e.  NN  |->  ( ( F `  ( 1st `  ( J `
 k ) ) ) `  ( 2nd `  ( J `  k
) ) ) )
5957, 58fmptd 6032 . . . . . . 7  |-  ( ph  ->  H : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
60 eqid 2402 . . . . . . . 8  |-  ( ( abs  o.  -  )  o.  H )  =  ( ( abs  o.  -  )  o.  H )
6160ovolfsval 22172 . . . . . . 7  |-  ( ( H : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  m  e.  NN )  ->  (
( ( abs  o.  -  )  o.  H
) `  m )  =  ( ( 2nd `  ( H `  m
) )  -  ( 1st `  ( H `  m ) ) ) )
6259, 12, 61syl2an 475 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( ( abs  o.  -  )  o.  H
) `  m )  =  ( ( 2nd `  ( H `  m
) )  -  ( 1st `  ( H `  m ) ) ) )
6312adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  m  e.  NN )
64 fveq2 5848 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( J `  k )  =  ( J `  m ) )
6564fveq2d 5852 . . . . . . . . . . . 12  |-  ( k  =  m  ->  ( 1st `  ( J `  k ) )  =  ( 1st `  ( J `  m )
) )
6665fveq2d 5852 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( F `  ( 1st `  ( J `  k
) ) )  =  ( F `  ( 1st `  ( J `  m ) ) ) )
6764fveq2d 5852 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( 2nd `  ( J `  k ) )  =  ( 2nd `  ( J `  m )
) )
6866, 67fveq12d 5854 . . . . . . . . . 10  |-  ( k  =  m  ->  (
( F `  ( 1st `  ( J `  k ) ) ) `
 ( 2nd `  ( J `  k )
) )  =  ( ( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )
69 fvex 5858 . . . . . . . . . 10  |-  ( ( F `  ( 1st `  ( J `  m
) ) ) `  ( 2nd `  ( J `
 m ) ) )  e.  _V
7068, 58, 69fvmpt 5931 . . . . . . . . 9  |-  ( m  e.  NN  ->  ( H `  m )  =  ( ( F `
 ( 1st `  ( J `  m )
) ) `  ( 2nd `  ( J `  m ) ) ) )
7163, 70syl 17 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( H `  m )  =  ( ( F `
 ( 1st `  ( J `  m )
) ) `  ( 2nd `  ( J `  m ) ) ) )
7271fveq2d 5852 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 2nd `  ( H `  m ) )  =  ( 2nd `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) ) )
7371fveq2d 5852 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 1st `  ( H `  m ) )  =  ( 1st `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) ) )
7472, 73oveq12d 6295 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( 2nd `  ( H `  m )
)  -  ( 1st `  ( H `  m
) ) )  =  ( ( 2nd `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )  -  ( 1st `  ( ( F `  ( 1st `  ( J `  m
) ) ) `  ( 2nd `  ( J `
 m ) ) ) ) ) )
7562, 74eqtrd 2443 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( ( abs  o.  -  )  o.  H
) `  m )  =  ( ( 2nd `  ( ( F `  ( 1st `  ( J `
 m ) ) ) `  ( 2nd `  ( J `  m
) ) ) )  -  ( 1st `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) ) ) )
76 ovoliun.k . . . . . 6  |-  ( ph  ->  K  e.  NN )
77 nnuz 11161 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
7876, 77syl6eleq 2500 . . . . 5  |-  ( ph  ->  K  e.  ( ZZ>= ` 
1 ) )
79 ffvelrn 6006 . . . . . . . . . . 11  |-  ( ( H : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  m  e.  NN )  ->  ( H `  m )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
8059, 12, 79syl2an 475 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( H `  m )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
8118, 80sseldi 3439 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( H `  m )  e.  ( RR  X.  RR ) )
82 xp2nd 6814 . . . . . . . . 9  |-  ( ( H `  m )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( H `  m
) )  e.  RR )
8381, 82syl 17 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 2nd `  ( H `  m ) )  e.  RR )
84 xp1st 6813 . . . . . . . . 9  |-  ( ( H `  m )  e.  ( RR  X.  RR )  ->  ( 1st `  ( H `  m
) )  e.  RR )
8581, 84syl 17 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 1st `  ( H `  m ) )  e.  RR )
8683, 85resubcld 10027 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( 2nd `  ( H `  m )
)  -  ( 1st `  ( H `  m
) ) )  e.  RR )
8786recnd 9651 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( 2nd `  ( H `  m )
)  -  ( 1st `  ( H `  m
) ) )  e.  CC )
8874, 87eqeltrrd 2491 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( 2nd `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )  -  ( 1st `  ( ( F `  ( 1st `  ( J `  m
) ) ) `  ( 2nd `  ( J `
 m ) ) ) ) )  e.  CC )
8975, 78, 88fsumser 13699 . . . 4  |-  ( ph  -> 
sum_ m  e.  (
1 ... K ) ( ( 2nd `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )  -  ( 1st `  ( ( F `  ( 1st `  ( J `  m
) ) ) `  ( 2nd `  ( J `
 m ) ) ) ) )  =  (  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  H ) ) `  K ) )
9047, 89eqtrd 2443 . . 3  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  =  (  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  H ) ) `  K ) )
91 ovoliun.u . . . 4  |-  U  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  H ) )
9291fveq1i 5849 . . 3  |-  ( U `
 K )  =  (  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  H ) ) `  K )
9390, 92syl6eqr 2461 . 2  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  =  ( U `  K
) )
94 f1oeng 7571 . . . . . . 7  |-  ( ( ( 1 ... K
)  e.  Fin  /\  ( J  |`  ( 1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " (
1 ... K ) ) )  ->  ( 1 ... K )  ~~  ( J " ( 1 ... K ) ) )
958, 15, 94syl2anc 659 . . . . . 6  |-  ( ph  ->  ( 1 ... K
)  ~~  ( J " ( 1 ... K
) ) )
9695ensymd 7603 . . . . 5  |-  ( ph  ->  ( J " (
1 ... K ) ) 
~~  ( 1 ... K ) )
97 enfii 7771 . . . . 5  |-  ( ( ( 1 ... K
)  e.  Fin  /\  ( J " ( 1 ... K ) ) 
~~  ( 1 ... K ) )  -> 
( J " (
1 ... K ) )  e.  Fin )
988, 96, 97syl2anc 659 . . . 4  |-  ( ph  ->  ( J " (
1 ... K ) )  e.  Fin )
9998, 45fsumrecl 13703 . . 3  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  e.  RR )
100 fzfid 12122 . . . . 5  |-  ( ph  ->  ( 1 ... L
)  e.  Fin )
101 elfznn 11766 . . . . . 6  |-  ( n  e.  ( 1 ... L )  ->  n  e.  NN )
102 ovoliun.v . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( vol* `  A )  e.  RR )
103101, 102sylan2 472 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol* `  A )  e.  RR )
104100, 103fsumrecl 13703 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  e.  RR )
105 ovoliun.b . . . . . . 7  |-  ( ph  ->  B  e.  RR+ )
106105rpred 11303 . . . . . 6  |-  ( ph  ->  B  e.  RR )
107 2nn 10733 . . . . . . . 8  |-  2  e.  NN
108 nnnn0 10842 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  NN0 )
109 nnexpcl 12221 . . . . . . . 8  |-  ( ( 2  e.  NN  /\  n  e.  NN0 )  -> 
( 2 ^ n
)  e.  NN )
110107, 108, 109sylancr 661 . . . . . . 7  |-  ( n  e.  NN  ->  (
2 ^ n )  e.  NN )
111101, 110syl 17 . . . . . 6  |-  ( n  e.  ( 1 ... L )  ->  (
2 ^ n )  e.  NN )
112 nndivre 10611 . . . . . 6  |-  ( ( B  e.  RR  /\  ( 2 ^ n
)  e.  NN )  ->  ( B  / 
( 2 ^ n
) )  e.  RR )
113106, 111, 112syl2an 475 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( B  /  ( 2 ^ n ) )  e.  RR )
114100, 113fsumrecl 13703 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  e.  RR )
115104, 114readdcld 9652 . . 3  |-  ( ph  ->  ( sum_ n  e.  ( 1 ... L ) ( vol* `  A )  +  sum_ n  e.  ( 1 ... L ) ( B  /  ( 2 ^ n ) ) )  e.  RR )
116 ovoliun.r . . . 4  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
117116, 106readdcld 9652 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  +  B )  e.  RR )
118 relxp 4930 . . . . . . . . . . . . . . 15  |-  Rel  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )
119 relres 5120 . . . . . . . . . . . . . . 15  |-  Rel  (
( J " (
1 ... K ) )  |`  { n } )
120 opelxp 4852 . . . . . . . . . . . . . . . 16  |-  ( <.
x ,  y >.  e.  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  <-> 
( x  e.  {
n }  /\  y  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )
121 vex 3061 . . . . . . . . . . . . . . . . . 18  |-  y  e. 
_V
122121opelres 5098 . . . . . . . . . . . . . . . . 17  |-  ( <.
x ,  y >.  e.  ( ( J "
( 1 ... K
) )  |`  { n } )  <->  ( <. x ,  y >.  e.  ( J " ( 1 ... K ) )  /\  x  e.  {
n } ) )
123 ancom 448 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  { n }  /\  <. x ,  y
>.  e.  ( J "
( 1 ... K
) ) )  <->  ( <. x ,  y >.  e.  ( J " ( 1 ... K ) )  /\  x  e.  {
n } ) )
124 elsni 3996 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  { n }  ->  x  =  n )
125124opeq1d 4164 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  { n }  -> 
<. x ,  y >.  =  <. n ,  y
>. )
126125eleq1d 2471 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  { n }  ->  ( <. x ,  y
>.  e.  ( J "
( 1 ... K
) )  <->  <. n ,  y >.  e.  ( J " ( 1 ... K ) ) ) )
127 vex 3061 . . . . . . . . . . . . . . . . . . . 20  |-  n  e. 
_V
128127, 121elimasn 5181 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( ( J
" ( 1 ... K ) ) " { n } )  <->  <. n ,  y >.  e.  ( J " (
1 ... K ) ) )
129126, 128syl6bbr 263 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  { n }  ->  ( <. x ,  y
>.  e.  ( J "
( 1 ... K
) )  <->  y  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )
130129pm5.32i 635 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  { n }  /\  <. x ,  y
>.  e.  ( J "
( 1 ... K
) ) )  <->  ( x  e.  { n }  /\  y  e.  ( ( J " ( 1 ... K ) ) " { n } ) ) )
131122, 123, 1303bitr2ri 274 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  { n }  /\  y  e.  ( ( J " (
1 ... K ) )
" { n }
) )  <->  <. x ,  y >.  e.  (
( J " (
1 ... K ) )  |`  { n } ) )
132120, 131bitri 249 . . . . . . . . . . . . . . 15  |-  ( <.
x ,  y >.  e.  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  <->  <. x ,  y >.  e.  ( ( J "
( 1 ... K
) )  |`  { n } ) )
133118, 119, 132eqrelriiv 4917 . . . . . . . . . . . . . 14  |-  ( { n }  X.  (
( J " (
1 ... K ) )
" { n }
) )  =  ( ( J " (
1 ... K ) )  |`  { n } )
134 df-res 4834 . . . . . . . . . . . . . 14  |-  ( ( J " ( 1 ... K ) )  |`  { n } )  =  ( ( J
" ( 1 ... K ) )  i^i  ( { n }  X.  _V ) )
135133, 134eqtri 2431 . . . . . . . . . . . . 13  |-  ( { n }  X.  (
( J " (
1 ... K ) )
" { n }
) )  =  ( ( J " (
1 ... K ) )  i^i  ( { n }  X.  _V ) )
136135a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  =  ( ( J
" ( 1 ... K ) )  i^i  ( { n }  X.  _V ) ) )
137136iuneq2dv 4292 . . . . . . . . . . 11  |-  ( ph  ->  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  =  U_ n  e.  ( 1 ... L
) ( ( J
" ( 1 ... K ) )  i^i  ( { n }  X.  _V ) ) )
138 iunin2 4334 . . . . . . . . . . 11  |-  U_ n  e.  ( 1 ... L
) ( ( J
" ( 1 ... K ) )  i^i  ( { n }  X.  _V ) )  =  ( ( J "
( 1 ... K
) )  i^i  U_ n  e.  ( 1 ... L ) ( { n }  X.  _V ) )
139137, 138syl6eq 2459 . . . . . . . . . 10  |-  ( ph  ->  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  =  ( ( J
" ( 1 ... K ) )  i^i  U_ n  e.  (
1 ... L ) ( { n }  X.  _V ) ) )
140 relxp 4930 . . . . . . . . . . . . . 14  |-  Rel  ( NN  X.  NN )
141 relss 4910 . . . . . . . . . . . . . 14  |-  ( ( J " ( 1 ... K ) ) 
C_  ( NN  X.  NN )  ->  ( Rel  ( NN  X.  NN )  ->  Rel  ( J " ( 1 ... K
) ) ) )
14226, 140, 141mpisyl 19 . . . . . . . . . . . . 13  |-  ( ph  ->  Rel  ( J "
( 1 ... K
) ) )
143 ovoliun.l2 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  A. w  e.  ( 1 ... K ) ( 1st `  ( J `  w )
)  <_  L )
144 ffn 5713 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J : NN --> ( NN 
X.  NN )  ->  J  Fn  NN )
14523, 144syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  J  Fn  NN )
146 fveq2 5848 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( j  =  ( J `  w )  ->  ( 1st `  j )  =  ( 1st `  ( J `  w )
) )
147146breq1d 4404 . . . . . . . . . . . . . . . . . . . . 21  |-  ( j  =  ( J `  w )  ->  (
( 1st `  j
)  <_  L  <->  ( 1st `  ( J `  w
) )  <_  L
) )
148147ralima 6132 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( J  Fn  NN  /\  ( 1 ... K
)  C_  NN )  ->  ( A. j  e.  ( J " (
1 ... K ) ) ( 1st `  j
)  <_  L  <->  A. w  e.  ( 1 ... K
) ( 1st `  ( J `  w )
)  <_  L )
)
149145, 13, 148sylancl 660 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( A. j  e.  ( J " (
1 ... K ) ) ( 1st `  j
)  <_  L  <->  A. w  e.  ( 1 ... K
) ( 1st `  ( J `  w )
)  <_  L )
)
150143, 149mpbird 232 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A. j  e.  ( J " ( 1 ... K ) ) ( 1st `  j
)  <_  L )
151150r19.21bi 2772 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  <_  L
)
15229, 77syl6eleq 2500 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  e.  (
ZZ>= `  1 ) )
153 ovoliun.l1 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  L  e.  ZZ )
154153adantr 463 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  L  e.  ZZ )
155 elfz5 11732 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  j
)  e.  ( ZZ>= ` 
1 )  /\  L  e.  ZZ )  ->  (
( 1st `  j
)  e.  ( 1 ... L )  <->  ( 1st `  j )  <_  L
) )
156152, 154, 155syl2anc 659 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( 1st `  j )  e.  ( 1 ... L
)  <->  ( 1st `  j
)  <_  L )
)
157151, 156mpbird 232 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  e.  ( 1 ... L ) )
158157ralrimiva 2817 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. j  e.  ( J " ( 1 ... K ) ) ( 1st `  j
)  e.  ( 1 ... L ) )
159 vex 3061 . . . . . . . . . . . . . . . . . 18  |-  x  e. 
_V
160159, 121op1std 6793 . . . . . . . . . . . . . . . . 17  |-  ( j  =  <. x ,  y
>.  ->  ( 1st `  j
)  =  x )
161160eleq1d 2471 . . . . . . . . . . . . . . . 16  |-  ( j  =  <. x ,  y
>.  ->  ( ( 1st `  j )  e.  ( 1 ... L )  <-> 
x  e.  ( 1 ... L ) ) )
162161rspccv 3156 . . . . . . . . . . . . . . 15  |-  ( A. j  e.  ( J " ( 1 ... K
) ) ( 1st `  j )  e.  ( 1 ... L )  ->  ( <. x ,  y >.  e.  ( J " ( 1 ... K ) )  ->  x  e.  ( 1 ... L ) ) )
163158, 162syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( J "
( 1 ... K
) )  ->  x  e.  ( 1 ... L
) ) )
164 opelxp 4852 . . . . . . . . . . . . . . 15  |-  ( <.
x ,  y >.  e.  ( U_ n  e.  ( 1 ... L
) { n }  X.  _V )  <->  ( x  e.  U_ n  e.  ( 1 ... L ) { n }  /\  y  e.  _V )
)
165121biantru 503 . . . . . . . . . . . . . . 15  |-  ( x  e.  U_ n  e.  ( 1 ... L
) { n }  <->  ( x  e.  U_ n  e.  ( 1 ... L
) { n }  /\  y  e.  _V ) )
166 iunid 4325 . . . . . . . . . . . . . . . 16  |-  U_ n  e.  ( 1 ... L
) { n }  =  ( 1 ... L )
167166eleq2i 2480 . . . . . . . . . . . . . . 15  |-  ( x  e.  U_ n  e.  ( 1 ... L
) { n }  <->  x  e.  ( 1 ... L ) )
168164, 165, 1673bitr2i 273 . . . . . . . . . . . . . 14  |-  ( <.
x ,  y >.  e.  ( U_ n  e.  ( 1 ... L
) { n }  X.  _V )  <->  x  e.  ( 1 ... L
) )
169163, 168syl6ibr 227 . . . . . . . . . . . . 13  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( J "
( 1 ... K
) )  ->  <. x ,  y >.  e.  (
U_ n  e.  ( 1 ... L ) { n }  X.  _V ) ) )
170142, 169relssdv 4915 . . . . . . . . . . . 12  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  ( U_ n  e.  ( 1 ... L
) { n }  X.  _V ) )
171 xpiundir 4878 . . . . . . . . . . . 12  |-  ( U_ n  e.  ( 1 ... L ) { n }  X.  _V )  =  U_ n  e.  ( 1 ... L
) ( { n }  X.  _V )
172170, 171syl6sseq 3487 . . . . . . . . . . 11  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  U_ n  e.  ( 1 ... L ) ( { n }  X.  _V ) )
173 df-ss 3427 . . . . . . . . . . 11  |-  ( ( J " ( 1 ... K ) ) 
C_  U_ n  e.  ( 1 ... L ) ( { n }  X.  _V )  <->  ( ( J " ( 1 ... K ) )  i^i  U_ n  e.  (
1 ... L ) ( { n }  X.  _V ) )  =  ( J " ( 1 ... K ) ) )
174172, 173sylib 196 . . . . . . . . . 10  |-  ( ph  ->  ( ( J "
( 1 ... K
) )  i^i  U_ n  e.  ( 1 ... L ) ( { n }  X.  _V ) )  =  ( J " ( 1 ... K ) ) )
175139, 174eqtrd 2443 . . . . . . . . 9  |-  ( ph  ->  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  =  ( J "
( 1 ... K
) ) )
176175, 98eqeltrd 2490 . . . . . . . 8  |-  ( ph  ->  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  e.  Fin )
177 ssiun2 4313 . . . . . . . 8  |-  ( n  e.  ( 1 ... L )  ->  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) 
C_  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )
178 ssfi 7774 . . . . . . . 8  |-  ( (
U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  e.  Fin  /\  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) 
C_  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( {
n }  X.  (
( J " (
1 ... K ) )
" { n }
) )  e.  Fin )
179176, 177, 178syl2an 475 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  e.  Fin )
180 2ndconst 6872 . . . . . . . . . 10  |-  ( n  e.  _V  ->  ( 2nd  |`  ( { n }  X.  ( ( J
" ( 1 ... K ) ) " { n } ) ) ) : ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) -1-1-onto-> ( ( J " (
1 ... K ) )
" { n }
) )
181127, 180ax-mp 5 . . . . . . . . 9  |-  ( 2nd  |`  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) ) : ( { n }  X.  (
( J " (
1 ... K ) )
" { n }
) ) -1-1-onto-> ( ( J "
( 1 ... K
) ) " {
n } )
182 f1oeng 7571 . . . . . . . . 9  |-  ( ( ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  e.  Fin  /\  ( 2nd  |`  ( { n }  X.  ( ( J
" ( 1 ... K ) ) " { n } ) ) ) : ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) -1-1-onto-> ( ( J " (
1 ... K ) )
" { n }
) )  ->  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) 
~~  ( ( J
" ( 1 ... K ) ) " { n } ) )
183179, 181, 182sylancl 660 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) 
~~  ( ( J
" ( 1 ... K ) ) " { n } ) )
184183ensymd 7603 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( J " (
1 ... K ) )
" { n }
)  ~~  ( {
n }  X.  (
( J " (
1 ... K ) )
" { n }
) ) )
185 enfii 7771 . . . . . . 7  |-  ( ( ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  e.  Fin  /\  (
( J " (
1 ... K ) )
" { n }
)  ~~  ( {
n }  X.  (
( J " (
1 ... K ) )
" { n }
) ) )  -> 
( ( J "
( 1 ... K
) ) " {
n } )  e. 
Fin )
186179, 184, 185syl2anc 659 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( J " (
1 ... K ) )
" { n }
)  e.  Fin )
187 ffvelrn 6006 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
18819, 101, 187syl2an 475 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( F `  n )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
18933, 34elmap 7484 . . . . . . . . . . . . 13  |-  ( ( F `  n )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  ( F `  n
) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
190188, 189sylib 196 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( F `  n ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
191190adantrr 715 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( F `  n ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
192 imassrn 5167 . . . . . . . . . . . . . 14  |-  ( ( J " ( 1 ... K ) )
" { n }
)  C_  ran  ( J
" ( 1 ... K ) )
19326adantr 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( J " ( 1 ... K ) )  C_  ( NN  X.  NN ) )
194 rnss 5051 . . . . . . . . . . . . . . . 16  |-  ( ( J " ( 1 ... K ) ) 
C_  ( NN  X.  NN )  ->  ran  ( J " ( 1 ... K ) )  C_  ran  ( NN  X.  NN ) )
195193, 194syl 17 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  ( J " ( 1 ... K ) ) 
C_  ran  ( NN  X.  NN ) )
196 rnxpid 5257 . . . . . . . . . . . . . . 15  |-  ran  ( NN  X.  NN )  =  NN
197195, 196syl6sseq 3487 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  ( J " ( 1 ... K ) ) 
C_  NN )
198192, 197syl5ss 3452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( J " (
1 ... K ) )
" { n }
)  C_  NN )
199198sseld 3440 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
i  e.  ( ( J " ( 1 ... K ) )
" { n }
)  ->  i  e.  NN ) )
200199impr 617 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  i  e.  NN )
201191, 200ffvelrnd 6009 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( F `  n ) `  i )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
20218, 201sseldi 3439 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( F `  n ) `  i )  e.  ( RR  X.  RR ) )
203 xp2nd 6814 . . . . . . . . 9  |-  ( ( ( F `  n
) `  i )  e.  ( RR  X.  RR )  ->  ( 2nd `  (
( F `  n
) `  i )
)  e.  RR )
204202, 203syl 17 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( 2nd `  ( ( F `  n ) `  i
) )  e.  RR )
205 xp1st 6813 . . . . . . . . 9  |-  ( ( ( F `  n
) `  i )  e.  ( RR  X.  RR )  ->  ( 1st `  (
( F `  n
) `  i )
)  e.  RR )
206202, 205syl 17 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( 1st `  ( ( F `  n ) `  i
) )  e.  RR )
207204, 206resubcld 10027 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( 2nd `  ( ( F `
 n ) `  i ) )  -  ( 1st `  ( ( F `  n ) `
 i ) ) )  e.  RR )
208207anassrs 646 . . . . . 6  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) )  ->  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  e.  RR )
209186, 208fsumrecl 13703 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  e.  RR )
210106, 110, 112syl2an 475 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( B  /  ( 2 ^ n ) )  e.  RR )
211102, 210readdcld 9652 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( vol* `  A
)  +  ( B  /  ( 2 ^ n ) ) )  e.  RR )
212101, 211sylan2 472 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( vol* `  A )  +  ( B  /  ( 2 ^ n ) ) )  e.  RR )
213 eqid 2402 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  o.  ( F `  n
) )  =  ( ( abs  o.  -  )  o.  ( F `  n ) )
214 ovoliun.s . . . . . . . . . . . 12  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  ( F `  n ) ) )
215213, 214ovolsf 22174 . . . . . . . . . . 11  |-  ( ( F `  n ) : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  S : NN --> ( 0 [,) +oo ) )
216190, 215syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  S : NN --> ( 0 [,) +oo ) )
217 frn 5719 . . . . . . . . . 10  |-  ( S : NN --> ( 0 [,) +oo )  ->  ran  S  C_  ( 0 [,) +oo ) )
218216, 217syl 17 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S 
C_  ( 0 [,) +oo ) )
219 icossxr 11661 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  RR*
220218, 219syl6ss 3453 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S 
C_  RR* )
221 supxrcl 11558 . . . . . . . 8  |-  ( ran 
S  C_  RR*  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
222220, 221syl 17 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
223 mnfxr 11375 . . . . . . . . 9  |- -oo  e.  RR*
224223a1i 11 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  -> -oo  e.  RR* )
225103rexrd 9672 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol* `  A )  e.  RR* )
226 mnflt 11385 . . . . . . . . 9  |-  ( ( vol* `  A
)  e.  RR  -> -oo 
<  ( vol* `  A ) )
227103, 226syl 17 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  -> -oo  <  ( vol* `  A
) )
228 ovoliun.x1 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  U.
ran  ( (,)  o.  ( F `  n ) ) )
229101, 228sylan2 472 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  A  C_ 
U. ran  ( (,)  o.  ( F `  n
) ) )
230214ovollb 22180 . . . . . . . . 9  |-  ( ( ( F `  n
) : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  ( F `  n
) ) )  -> 
( vol* `  A )  <_  sup ( ran  S ,  RR* ,  <  ) )
231190, 229, 230syl2anc 659 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol* `  A )  <_  sup ( ran  S ,  RR* ,  <  )
)
232224, 225, 222, 227, 231xrltletrd 11416 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  -> -oo  <  sup ( ran  S ,  RR* ,  <  ) )
233 ovoliun.x2 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  sup ( ran  S ,  RR* ,  <  )  <_  ( ( vol* `  A )  +  ( B  / 
( 2 ^ n
) ) ) )
234101, 233sylan2 472 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  <_  (
( vol* `  A )  +  ( B  /  ( 2 ^ n ) ) ) )
235 xrre 11422 . . . . . . 7  |-  ( ( ( sup ( ran 
S ,  RR* ,  <  )  e.  RR*  /\  (
( vol* `  A )  +  ( B  /  ( 2 ^ n ) ) )  e.  RR )  /\  ( -oo  <  sup ( ran  S ,  RR* ,  <  )  /\  sup ( ran  S ,  RR* ,  <  )  <_ 
( ( vol* `  A )  +  ( B  /  ( 2 ^ n ) ) ) ) )  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR )
236222, 212, 232, 234, 235syl22anc 1231 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR )
237 1zzd 10935 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  1  e.  ZZ )
238213ovolfsval 22172 . . . . . . . . 9  |-  ( ( ( F `  n
) : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  i  e.  NN )  ->  (
( ( abs  o.  -  )  o.  ( F `  n )
) `  i )  =  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) ) )
239190, 238sylan 469 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( ( abs  o.  -  )  o.  ( F `  n )
) `  i )  =  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) ) )
240213ovolfsf 22173 . . . . . . . . . . . . 13  |-  ( ( F `  n ) : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  ( F `  n
) ) : NN --> ( 0 [,) +oo ) )
241190, 240syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( abs  o.  -  )  o.  ( F `  n
) ) : NN --> ( 0 [,) +oo ) )
242241ffvelrnda 6008 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( ( abs  o.  -  )  o.  ( F `  n )
) `  i )  e.  ( 0 [,) +oo ) )
243239, 242eqeltrrd 2491 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  e.  ( 0 [,) +oo ) )
244 elrege0 11679 . . . . . . . . . 10  |-  ( ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  e.  ( 0 [,) +oo ) 
<->  ( ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  e.  RR  /\  0  <_  ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) ) ) )
245243, 244sylib 196 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  e.  RR  /\  0  <_ 
( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) ) ) )
246245simpld 457 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  e.  RR )
247245simprd 461 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  0  <_  ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) ) )
248 supxrub 11568 . . . . . . . . . . . . . . 15  |-  ( ( ran  S  C_  RR*  /\  z  e.  ran  S )  -> 
z  <_  sup ( ran  S ,  RR* ,  <  ) )
249220, 248sylan 469 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  z  e.  ran  S )  -> 
z  <_  sup ( ran  S ,  RR* ,  <  ) )
250249ralrimiva 2817 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  A. z  e.  ran  S  z  <_  sup ( ran  S ,  RR* ,  <  ) )
251 breq2 4398 . . . . . . . . . . . . . . 15  |-  ( x  =  sup ( ran 
S ,  RR* ,  <  )  ->  ( z  <_  x 
<->  z  <_  sup ( ran  S ,  RR* ,  <  ) ) )
252251ralbidv 2842 . . . . . . . . . . . . . 14  |-  ( x  =  sup ( ran 
S ,  RR* ,  <  )  ->  ( A. z  e.  ran  S  z  <_  x 
<-> 
A. z  e.  ran  S  z  <_  sup ( ran  S ,  RR* ,  <  ) ) )
253252rspcev 3159 . . . . . . . . . . . . 13  |-  ( ( sup ( ran  S ,  RR* ,  <  )  e.  RR  /\  A. z  e.  ran  S  z  <_  sup ( ran  S ,  RR* ,  <  ) )  ->  E. x  e.  RR  A. z  e.  ran  S  z  <_  x )
254236, 250, 253syl2anc 659 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  E. x  e.  RR  A. z  e. 
ran  S  z  <_  x )
255 ffn 5713 . . . . . . . . . . . . . . 15  |-  ( S : NN --> ( 0 [,) +oo )  ->  S  Fn  NN )
256216, 255syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  S  Fn  NN )
257 breq1 4397 . . . . . . . . . . . . . . 15  |-  ( z  =  ( S `  k )  ->  (
z  <_  x  <->  ( S `  k )  <_  x
) )
258257ralrn 6011 . . . . . . . . . . . . . 14  |-  ( S  Fn  NN  ->  ( A. z  e.  ran  S  z  <_  x  <->  A. k  e.  NN  ( S `  k )  <_  x
) )
259256, 258syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( A. z  e.  ran  S  z  <_  x  <->  A. k  e.  NN  ( S `  k )  <_  x
) )
260259rexbidv 2917 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( E. x  e.  RR  A. z  e.  ran  S  z  <_  x  <->  E. x  e.  RR  A. k  e.  NN  ( S `  k )  <_  x
) )
261254, 260mpbid 210 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  E. x  e.  RR  A. k  e.  NN  ( S `  k )  <_  x
)
26277, 214, 237, 239, 246, 247, 261isumsup2 13807 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  S  ~~>  sup ( ran  S ,  RR ,  <  ) )
263214, 262syl5eqbrr 4428 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  ( F `  n )
) )  ~~>  sup ( ran  S ,  RR ,  <  ) )
264 climrel 13462 . . . . . . . . . 10  |-  Rel  ~~>
265264releldmi 5059 . . . . . . . . 9  |-  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  ( F `  n )
) )  ~~>  sup ( ran  S ,  RR ,  <  )  ->  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  ( F `  n ) ) )  e.  dom  ~~>  )
266263, 265syl 17 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  ( F `  n )
) )  e.  dom  ~~>  )
26777, 237, 186, 198, 239, 246, 247, 266isumless 13806 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  <_  sum_ i  e.  NN  (
( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) ) )
26877, 214, 237, 239, 246, 247, 261isumsup 13808 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  NN  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  =  sup ( ran 
S ,  RR ,  <  ) )
269 rge0ssre 11680 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
270218, 269syl6ss 3453 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S 
C_  RR )
271 1nn 10586 . . . . . . . . . . . 12  |-  1  e.  NN
272 fdm 5717 . . . . . . . . . . . . 13  |-  ( S : NN --> ( 0 [,) +oo )  ->  dom  S  =  NN )
273216, 272syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  dom  S  =  NN )
274271, 273syl5eleqr 2497 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  1  e.  dom  S )
275 ne0i 3743 . . . . . . . . . . 11  |-  ( 1  e.  dom  S  ->  dom  S  =/=  (/) )
276274, 275syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  dom  S  =/=  (/) )
277 dm0rn0 5039 . . . . . . . . . . 11  |-  ( dom 
S  =  (/)  <->  ran  S  =  (/) )
278277necon3bii 2671 . . . . . . . . . 10  |-  ( dom 
S  =/=  (/)  <->  ran  S  =/=  (/) )
279276, 278sylib 196 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S  =/=  (/) )
280 supxrre 11571 . . . . . . . . 9  |-  ( ( ran  S  C_  RR  /\ 
ran  S  =/=  (/)  /\  E. x  e.  RR  A. z  e.  ran  S  z  <_  x )  ->  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  S ,  RR ,  <  ) )
281270, 279, 254, 280syl3anc 1230 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  S ,  RR ,  <  ) )
282268, 281eqtr4d 2446 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  NN  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  =  sup ( ran 
S ,  RR* ,  <  ) )
283267, 282breqtrd 4418 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  <_  sup ( ran  S ,  RR* ,  <  ) )
284209, 236, 212, 283, 234letrd 9772 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  <_ 
( ( vol* `  A )  +  ( B  /  ( 2 ^ n ) ) ) )
285100, 209, 212, 284fsumle 13762 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) sum_ i  e.  ( ( J " ( 1 ... K ) ) " { n } ) ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  <_  sum_ n  e.  ( 1 ... L ) ( ( vol* `  A )  +  ( B  /  ( 2 ^ n ) ) ) )
286 vex 3061 . . . . . . . . . . 11  |-  i  e. 
_V
287127, 286op1std 6793 . . . . . . . . . 10  |-  ( j  =  <. n ,  i
>.  ->  ( 1st `  j
)  =  n )
288287fveq2d 5852 . . . . . . . . 9  |-  ( j  =  <. n ,  i
>.  ->  ( F `  ( 1st `  j ) )  =  ( F `
 n ) )
289127, 286op2ndd 6794 . . . . . . . . 9  |-  ( j  =  <. n ,  i
>.  ->  ( 2nd `  j
)  =  i )
290288, 289fveq12d 5854 . . . . . . . 8  |-  ( j  =  <. n ,  i
>.  ->  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) )  =  ( ( F `
 n ) `  i ) )
291290fveq2d 5852 . . . . . . 7  |-  ( j  =  <. n ,  i
>.  ->  ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  =  ( 2nd `  (
( F `  n
) `  i )
) )
292290fveq2d 5852 . . . . . . 7  |-  ( j  =  <. n ,  i
>.  ->  ( 1st `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  =  ( 1st `  (
( F `  n
) `  i )
) )
293291, 292oveq12d 6295 . . . . . 6  |-  ( j  =  <. n ,  i
>.  ->  ( ( 2nd `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) )  -  ( 1st `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) ) )  =  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) ) )
294207recnd 9651 . . . . . 6  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( 2nd `  ( ( F `
 n ) `  i ) )  -  ( 1st `  ( ( F `  n ) `
 i ) ) )  e.  CC )
295293, 100, 186, 294fsum2d 13735 . . . . 5  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) sum_ i  e.  ( ( J " ( 1 ... K ) ) " { n } ) ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  = 
sum_ j  e.  U_  n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) ) )
296175sumeq1d 13670 . . . . 5  |-  ( ph  -> 
sum_ j  e.  U_  n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  = 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) ) )
297295, 296eqtrd 2443 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) sum_ i  e.  ( ( J " ( 1 ... K ) ) " { n } ) ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  = 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) ) )
298103recnd 9651 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol* `  A )  e.  CC )
299113recnd 9651 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( B  /  ( 2 ^ n ) )  e.  CC )
300100, 298, 299fsumadd 13708 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( ( vol* `  A )  +  ( B  /  ( 2 ^ n ) ) )  =  ( sum_ n  e.  ( 1 ... L ) ( vol* `  A )  +  sum_ n  e.  ( 1 ... L ) ( B  /  (
2 ^ n ) ) ) )
301285, 297, 3003brtr3d 4423 . . 3  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  <_ 
( sum_ n  e.  ( 1 ... L ) ( vol* `  A )  +  sum_ n  e.  ( 1 ... L ) ( B  /  ( 2 ^ n ) ) ) )
302 1zzd 10935 . . . . . . . . 9  |-  ( ph  ->  1  e.  ZZ )
303 simpr 459 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  NN )
304 ovoliun.g . . . . . . . . . . . 12  |-  G  =  ( n  e.  NN  |->  ( vol* `  A
) )
305304fvmpt2 5940 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  ( vol* `  A
)  e.  RR )  ->  ( G `  n )  =  ( vol* `  A
) )
306303, 102, 305syl2anc 659 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  =  ( vol* `  A ) )
307306, 102eqeltrd 2490 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
30877, 302, 307serfre 12178 . . . . . . . 8  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> RR )
309 ovoliun.t . . . . . . . . 9  |-  T  =  seq 1 (  +  ,  G )
310309feq1i 5705 . . . . . . . 8  |-  ( T : NN --> RR  <->  seq 1
(  +  ,  G
) : NN --> RR )
311308, 310sylibr 212 . . . . . . 7  |-  ( ph  ->  T : NN --> RR )
312 frn 5719 . . . . . . 7  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
313311, 312syl 17 . . . . . 6  |-  ( ph  ->  ran  T  C_  RR )
314 ressxr 9666 . . . . . 6  |-  RR  C_  RR*
315313, 314syl6ss 3453 . . . . 5  |-  ( ph  ->  ran  T  C_  RR* )
316101, 306sylan2 472 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( G `  n )  =  ( vol* `  A ) )
317 1red 9640 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
318 ffvelrn 6006 . . . . . . . . . . . . 13  |-  ( ( J : NN --> ( NN 
X.  NN )  /\  1  e.  NN )  ->  ( J `  1
)  e.  ( NN 
X.  NN ) )
31923, 271, 318sylancl 660 . . . . . . . . . . . 12  |-  ( ph  ->  ( J `  1
)  e.  ( NN 
X.  NN ) )
320 xp1st 6813 . . . . . . . . . . . 12  |-  ( ( J `  1 )  e.  ( NN  X.  NN )  ->  ( 1st `  ( J `  1
) )  e.  NN )
321319, 320syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  e.  NN )
322321nnred 10590 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  e.  RR )
323153zred 11007 . . . . . . . . . 10  |-  ( ph  ->  L  e.  RR )
324321nnge1d 10618 . . . . . . . . . 10  |-  ( ph  ->  1  <_  ( 1st `  ( J `  1
) ) )
325 eluzfz1 11745 . . . . . . . . . . . 12  |-  ( K  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... K
) )
32678, 325syl 17 . . . . . . . . . . 11  |-  ( ph  ->  1  e.  ( 1 ... K ) )
327 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( w  =  1  ->  ( J `  w )  =  ( J ` 
1 ) )
328327fveq2d 5852 . . . . . . . . . . . . 13  |-  ( w  =  1  ->  ( 1st `  ( J `  w ) )  =  ( 1st `  ( J `  1 )
) )
329328breq1d 4404 . . . . . . . . . . . 12  |-  ( w  =  1  ->  (
( 1st `  ( J `  w )
)  <_  L  <->  ( 1st `  ( J `  1
) )  <_  L
) )
330329rspcv 3155 . . . . . . . . . . 11  |-  ( 1  e.  ( 1 ... K )  ->  ( A. w  e.  (
1 ... K ) ( 1st `  ( J `
 w ) )  <_  L  ->  ( 1st `  ( J ` 
1 ) )  <_  L ) )
331326, 143, 330sylc 59 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  <_  L )
332317, 322, 323, 324, 331letrd 9772 . . . . . . . . 9  |-  ( ph  ->  1  <_  L )
333 elnnz1 10930 . . . . . . . . 9  |-  ( L  e.  NN  <->  ( L  e.  ZZ  /\  1  <_  L ) )
334153, 332, 333sylanbrc 662 . . . . . . . 8  |-  ( ph  ->  L  e.  NN )
335334, 77syl6eleq 2500 . . . . . . 7  |-  ( ph  ->  L  e.  ( ZZ>= ` 
1 ) )
336316, 335, 298fsumser 13699 . . . . . 6  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  =  (  seq 1 (  +  ,  G ) `  L
) )
337 seqfn 12161 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  seq 1 (  +  ,  G )  Fn  ( ZZ>=
`  1 ) )
338302, 337syl 17 . . . . . . . 8  |-  ( ph  ->  seq 1 (  +  ,  G )  Fn  ( ZZ>= `  1 )
)
339 fnfvelrn 6005 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  G )  Fn  ( ZZ>= `  1 )  /\  L  e.  ( ZZ>=
`  1 ) )  ->  (  seq 1
(  +  ,  G
) `  L )  e.  ran  seq 1 (  +  ,  G ) )
340338, 335, 339syl2anc 659 . . . . . . 7  |-  ( ph  ->  (  seq 1 (  +  ,  G ) `
 L )  e. 
ran  seq 1 (  +  ,  G ) )
341309rneqi 5049 . . . . . . 7  |-  ran  T  =  ran  seq 1 (  +  ,  G )
342340, 341syl6eleqr 2501 . . . . . 6  |-  ( ph  ->  (  seq 1 (  +  ,  G ) `
 L )  e. 
ran  T )
343336, 342eqeltrd 2490 . . . . 5  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  e.  ran  T
)
344 supxrub 11568 . . . . 5  |-  ( ( ran  T  C_  RR*  /\  sum_ n  e.  ( 1 ... L ) ( vol* `  A )  e.  ran  T )  ->  sum_ n  e.  ( 1 ... L ) ( vol* `  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
345315, 343, 344syl2anc 659 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
346106recnd 9651 . . . . . 6  |-  ( ph  ->  B  e.  CC )
347 geo2sum 13832 . . . . . 6  |-  ( ( L  e.  NN  /\  B  e.  CC )  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  =  ( B  -  ( B  /  (
2 ^ L ) ) ) )
348334, 346, 347syl2anc 659 . . . . 5  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  =  ( B  -  ( B  /  (
2 ^ L ) ) ) )
349334nnnn0d 10892 . . . . . . . . . 10  |-  ( ph  ->  L  e.  NN0 )
350 nnexpcl 12221 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  L  e.  NN0 )  -> 
( 2 ^ L
)  e.  NN )
351107, 349, 350sylancr 661 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ L
)  e.  NN )
352351nnrpd 11301 . . . . . . . 8  |-  ( ph  ->  ( 2 ^ L
)  e.  RR+ )
353105, 352rpdivcld 11320 . . . . . . 7  |-  ( ph  ->  ( B  /  (
2 ^ L ) )  e.  RR+ )
354353rpge0d 11307 . . . . . 6  |-  ( ph  ->  0  <_  ( B  /  ( 2 ^ L ) ) )
355106, 351nndivred 10624 . . . . . . 7  |-  ( ph  ->  ( B  /  (
2 ^ L ) )  e.  RR )
356106, 355subge02d 10183 . . . . . 6  |-  ( ph  ->  ( 0  <_  ( B  /  ( 2 ^ L ) )  <->  ( B  -  ( B  / 
( 2 ^ L
) ) )  <_  B ) )
357354, 356mpbid 210 . . . . 5  |-  ( ph  ->  ( B  -  ( B  /  ( 2 ^ L ) ) )  <_  B )
358348, 357eqbrtrd 4414 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  <_  B )
359104, 114, 116, 106, 345, 358le2addd 10209 . . 3  |-  ( ph  ->  ( sum_ n  e.  ( 1 ... L ) ( vol* `  A )  +  sum_ n  e.  ( 1 ... L ) ( B  /  ( 2 ^ n ) ) )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  B ) )
36099, 115, 117, 301, 359letrd 9772 . 2  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  <_ 
( sup ( ran 
T ,  RR* ,  <  )  +  B ) )
36193, 360eqbrtrrd 4416 1  |-  ( ph  ->  ( U `  K
)  <_  ( sup ( ran  T ,  RR* ,  <  )  +  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754   _Vcvv 3058    i^i cin 3412    C_ wss 3413   (/)c0 3737   {csn 3971   <.cop 3977   U.cuni 4190   U_ciun 4270   class class class wbr 4394    |-> cmpt 4452    X. cxp 4820   dom cdm 4822   ran crn 4823    |` cres 4824   "cima 4825    o. ccom 4826   Rel wrel 4827    Fn wfn 5563   -->wf 5564   -1-1->wf1 5565   -1-1-onto->wf1o 5567   ` cfv 5568  (class class class)co 6277   1stc1st 6781   2ndc2nd 6782    ^m cmap 7456    ~~ cen 7550   Fincfn 7553   supcsup 7933   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522    + caddc 9524   +oocpnf 9654   -oocmnf 9655   RR*cxr 9656    < clt 9657    <_ cle 9658    - cmin 9840    / cdiv 10246   NNcn 10575   2c2 10625   NN0cn0 10835   ZZcz 10904   ZZ>=cuz 11126   RR+crp 11264   (,)cioo 11581   [,)cico 11583   ...cfz 11724    seqcseq 12149   ^cexp 12208   abscabs 13214    ~~> cli 13454   sum_csu 13655   vol*covol 22164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-ioo 11585  df-ico 11587  df-fz 11725  df-fzo 11853  df-fl 11964  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-rlim 13459  df-sum 13656  df-ovol 22166
This theorem is referenced by:  ovoliunlem2  22204
  Copyright terms: Public domain W3C validator