MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovoliunlem1 Unicode version

Theorem ovoliunlem1 18861
Description: Lemma for ovoliun 18864. (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 5525 . . . . . . . . 9  |-  ( j  =  ( J `  m )  ->  ( 1st `  j )  =  ( 1st `  ( J `  m )
) )
21fveq2d 5529 . . . . . . . 8  |-  ( j  =  ( J `  m )  ->  ( F `  ( 1st `  j ) )  =  ( F `  ( 1st `  ( J `  m ) ) ) )
3 fveq2 5525 . . . . . . . 8  |-  ( j  =  ( J `  m )  ->  ( 2nd `  j )  =  ( 2nd `  ( J `  m )
) )
42, 3fveq12d 5531 . . . . . . 7  |-  ( j  =  ( J `  m )  ->  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) )  =  ( ( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )
54fveq2d 5529 . . . . . 6  |-  ( j  =  ( J `  m )  ->  ( 2nd `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  =  ( 2nd `  ( ( F `  ( 1st `  ( J `
 m ) ) ) `  ( 2nd `  ( J `  m
) ) ) ) )
64fveq2d 5529 . . . . . 6  |-  ( j  =  ( J `  m )  ->  ( 1st `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  =  ( 1st `  ( ( F `  ( 1st `  ( J `
 m ) ) ) `  ( 2nd `  ( J `  m
) ) ) ) )
75, 6oveq12d 5876 . . . . 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 11035 . . . . 5  |-  ( ph  ->  ( 1 ... K
)  e.  Fin )
9 ovoliun.j . . . . . . 7  |-  ( ph  ->  J : NN -1-1-onto-> ( NN  X.  NN ) )
10 f1of1 5471 . . . . . . 7  |-  ( J : NN -1-1-onto-> ( NN  X.  NN )  ->  J : NN -1-1-> ( NN  X.  NN ) )
119, 10syl 15 . . . . . 6  |-  ( ph  ->  J : NN -1-1-> ( NN  X.  NN ) )
12 elfznn 10819 . . . . . . 7  |-  ( m  e.  ( 1 ... K )  ->  m  e.  NN )
1312ssriv 3184 . . . . . 6  |-  ( 1 ... K )  C_  NN
14 f1ores 5487 . . . . . 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 643 . . . . 5  |-  ( ph  ->  ( J  |`  (
1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " (
1 ... K ) ) )
16 fvres 5542 . . . . . 6  |-  ( m  e.  ( 1 ... K )  ->  (
( J  |`  (
1 ... K ) ) `
 m )  =  ( J `  m
) )
1716adantl 452 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( J  |`  (
1 ... K ) ) `
 m )  =  ( J `  m
) )
18 inss2 3390 . . . . . . . . 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 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  F : NN
--> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
21 imassrn 5025 . . . . . . . . . . . . . . 15  |-  ( J
" ( 1 ... K ) )  C_  ran  J
22 f1of 5472 . . . . . . . . . . . . . . . . 17  |-  ( J : NN -1-1-onto-> ( NN  X.  NN )  ->  J : NN --> ( NN  X.  NN ) )
239, 22syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  J : NN --> ( NN 
X.  NN ) )
24 frn 5395 . . . . . . . . . . . . . . . 16  |-  ( J : NN --> ( NN 
X.  NN )  ->  ran  J  C_  ( NN  X.  NN ) )
2523, 24syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  J  C_  ( NN  X.  NN ) )
2621, 25syl5ss 3190 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  ( NN  X.  NN ) )
2726sselda 3180 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  j  e.  ( NN  X.  NN ) )
28 xp1st 6149 . . . . . . . . . . . . 13  |-  ( j  e.  ( NN  X.  NN )  ->  ( 1st `  j )  e.  NN )
2927, 28syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  e.  NN )
30 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( F : NN --> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( 1st `  j )  e.  NN )  ->  ( F `  ( 1st `  j ) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
3120, 29, 30syl2anc 642 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( F `  ( 1st `  j
) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
32 reex 8828 . . . . . . . . . . . . . 14  |-  RR  e.  _V
3332, 32xpex 4801 . . . . . . . . . . . . 13  |-  ( RR 
X.  RR )  e. 
_V
3433inex2 4156 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
35 nnex 9752 . . . . . . . . . . . 12  |-  NN  e.  _V
3634, 35elmap 6796 . . . . . . . . . . 11  |-  ( ( F `  ( 1st `  j ) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  ( F `  ( 1st `  j ) ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
3731, 36sylib 188 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( F `  ( 1st `  j
) ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
38 xp2nd 6150 . . . . . . . . . . 11  |-  ( j  e.  ( NN  X.  NN )  ->  ( 2nd `  j )  e.  NN )
3927, 38syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 2nd `  j )  e.  NN )
40 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( ( F `  ( 1st `  j ) ) : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  ( 2nd `  j )  e.  NN )  ->  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
4137, 39, 40syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
4218, 41sseldi 3178 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) )  e.  ( RR 
X.  RR ) )
43 xp2nd 6150 . . . . . . . 8  |-  ( ( ( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) )  e.  ( RR  X.  RR )  ->  ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  e.  RR )
4442, 43syl 15 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 2nd `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) )  e.  RR )
45 xp1st 6149 . . . . . . . 8  |-  ( ( ( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) )  e.  ( RR  X.  RR )  ->  ( 1st `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  e.  RR )
4642, 45syl 15 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) )  e.  RR )
4744, 46resubcld 9211 . . . . . 6  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( 2nd `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  e.  RR )
4847recnd 8861 . . . . 5  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( 2nd `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  e.  CC )
497, 8, 15, 17, 48fsumf1o 12196 . . . 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 ) ) ) ) ) )
5019adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  F : NN
--> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
51 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( J : NN --> ( NN 
X.  NN )  /\  k  e.  NN )  ->  ( J `  k
)  e.  ( NN 
X.  NN ) )
5223, 51sylan 457 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  ( J `
 k )  e.  ( NN  X.  NN ) )
53 xp1st 6149 . . . . . . . . . . . 12  |-  ( ( J `  k )  e.  ( NN  X.  NN )  ->  ( 1st `  ( J `  k
) )  e.  NN )
5452, 53syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1st `  ( J `  k
) )  e.  NN )
55 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( F : NN --> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  ( 1st `  ( J `  k ) )  e.  NN )  ->  ( F `  ( 1st `  ( J `  k
) ) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
5650, 54, 55syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 ( 1st `  ( J `  k )
) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
5734, 35elmap 6796 . . . . . . . . . 10  |-  ( ( F `  ( 1st `  ( J `  k
) ) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  ( F `  ( 1st `  ( J `  k ) ) ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
5856, 57sylib 188 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 ( 1st `  ( J `  k )
) ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
59 xp2nd 6150 . . . . . . . . . 10  |-  ( ( J `  k )  e.  ( NN  X.  NN )  ->  ( 2nd `  ( J `  k
) )  e.  NN )
6052, 59syl 15 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( 2nd `  ( J `  k
) )  e.  NN )
61 ffvelrn 5663 . . . . . . . . 9  |-  ( ( ( F `  ( 1st `  ( J `  k ) ) ) : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  ( 2nd `  ( J `  k ) )  e.  NN )  ->  (
( F `  ( 1st `  ( J `  k ) ) ) `
 ( 2nd `  ( J `  k )
) )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
6258, 60, 61syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( F `  ( 1st `  ( J `  k
) ) ) `  ( 2nd `  ( J `
 k ) ) )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
63 ovoliun.h . . . . . . . 8  |-  H  =  ( k  e.  NN  |->  ( ( F `  ( 1st `  ( J `
 k ) ) ) `  ( 2nd `  ( J `  k
) ) ) )
6462, 63fmptd 5684 . . . . . . 7  |-  ( ph  ->  H : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
65 eqid 2283 . . . . . . . 8  |-  ( ( abs  o.  -  )  o.  H )  =  ( ( abs  o.  -  )  o.  H )
6665ovolfsval 18830 . . . . . . 7  |-  ( ( H : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  m  e.  NN )  ->  (
( ( abs  o.  -  )  o.  H
) `  m )  =  ( ( 2nd `  ( H `  m
) )  -  ( 1st `  ( H `  m ) ) ) )
6764, 12, 66syl2an 463 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( ( abs  o.  -  )  o.  H
) `  m )  =  ( ( 2nd `  ( H `  m
) )  -  ( 1st `  ( H `  m ) ) ) )
6812adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  m  e.  NN )
69 fveq2 5525 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( J `  k )  =  ( J `  m ) )
7069fveq2d 5529 . . . . . . . . . . . 12  |-  ( k  =  m  ->  ( 1st `  ( J `  k ) )  =  ( 1st `  ( J `  m )
) )
7170fveq2d 5529 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( F `  ( 1st `  ( J `  k
) ) )  =  ( F `  ( 1st `  ( J `  m ) ) ) )
7269fveq2d 5529 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( 2nd `  ( J `  k ) )  =  ( 2nd `  ( J `  m )
) )
7371, 72fveq12d 5531 . . . . . . . . . 10  |-  ( k  =  m  ->  (
( F `  ( 1st `  ( J `  k ) ) ) `
 ( 2nd `  ( J `  k )
) )  =  ( ( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )
74 fvex 5539 . . . . . . . . . 10  |-  ( ( F `  ( 1st `  ( J `  m
) ) ) `  ( 2nd `  ( J `
 m ) ) )  e.  _V
7573, 63, 74fvmpt 5602 . . . . . . . . 9  |-  ( m  e.  NN  ->  ( H `  m )  =  ( ( F `
 ( 1st `  ( J `  m )
) ) `  ( 2nd `  ( J `  m ) ) ) )
7668, 75syl 15 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( H `  m )  =  ( ( F `
 ( 1st `  ( J `  m )
) ) `  ( 2nd `  ( J `  m ) ) ) )
7776fveq2d 5529 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 2nd `  ( H `  m ) )  =  ( 2nd `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) ) )
7876fveq2d 5529 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 1st `  ( H `  m ) )  =  ( 1st `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) ) )
7977, 78oveq12d 5876 . . . . . 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 ) ) ) ) ) )
8067, 79eqtrd 2315 . . . . 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 )
) ) ) ) )
81 ovoliun.k . . . . . 6  |-  ( ph  ->  K  e.  NN )
82 nnuz 10263 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
8381, 82syl6eleq 2373 . . . . 5  |-  ( ph  ->  K  e.  ( ZZ>= ` 
1 ) )
84 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( H : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  m  e.  NN )  ->  ( H `  m )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
8564, 12, 84syl2an 463 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( H `  m )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
8618, 85sseldi 3178 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( H `  m )  e.  ( RR  X.  RR ) )
87 xp2nd 6150 . . . . . . . . 9  |-  ( ( H `  m )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( H `  m
) )  e.  RR )
8886, 87syl 15 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 2nd `  ( H `  m ) )  e.  RR )
89 xp1st 6149 . . . . . . . . 9  |-  ( ( H `  m )  e.  ( RR  X.  RR )  ->  ( 1st `  ( H `  m
) )  e.  RR )
9086, 89syl 15 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 1st `  ( H `  m ) )  e.  RR )
9188, 90resubcld 9211 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( 2nd `  ( H `  m )
)  -  ( 1st `  ( H `  m
) ) )  e.  RR )
9291recnd 8861 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( 2nd `  ( H `  m )
)  -  ( 1st `  ( H `  m
) ) )  e.  CC )
9379, 92eqeltrrd 2358 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( 2nd `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )  -  ( 1st `  ( ( F `  ( 1st `  ( J `  m
) ) ) `  ( 2nd `  ( J `
 m ) ) ) ) )  e.  CC )
9480, 83, 93fsumser 12203 . . . 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 ) )
9549, 94eqtrd 2315 . . 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 ) )
96 ovoliun.u . . . 4  |-  U  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  H ) )
9796fveq1i 5526 . . 3  |-  ( U `
 K )  =  (  seq  1 (  +  ,  ( ( abs  o.  -  )  o.  H ) ) `  K )
9895, 97syl6eqr 2333 . 2  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  =  ( U `  K
) )
99 f1oeng 6880 . . . . . . 7  |-  ( ( ( 1 ... K
)  e.  Fin  /\  ( J  |`  ( 1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " (
1 ... K ) ) )  ->  ( 1 ... K )  ~~  ( J " ( 1 ... K ) ) )
1008, 15, 99syl2anc 642 . . . . . 6  |-  ( ph  ->  ( 1 ... K
)  ~~  ( J " ( 1 ... K
) ) )
101 ensym 6910 . . . . . 6  |-  ( ( 1 ... K ) 
~~  ( J "
( 1 ... K
) )  ->  ( J " ( 1 ... K ) )  ~~  ( 1 ... K
) )
102100, 101syl 15 . . . . 5  |-  ( ph  ->  ( J " (
1 ... K ) ) 
~~  ( 1 ... K ) )
103 enfii 7080 . . . . 5  |-  ( ( ( 1 ... K
)  e.  Fin  /\  ( J " ( 1 ... K ) ) 
~~  ( 1 ... K ) )  -> 
( J " (
1 ... K ) )  e.  Fin )
1048, 102, 103syl2anc 642 . . . 4  |-  ( ph  ->  ( J " (
1 ... K ) )  e.  Fin )
105104, 47fsumrecl 12207 . . 3  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  e.  RR )
106 fzfid 11035 . . . . 5  |-  ( ph  ->  ( 1 ... L
)  e.  Fin )
107 elfznn 10819 . . . . . 6  |-  ( n  e.  ( 1 ... L )  ->  n  e.  NN )
108 ovoliun.v . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( vol
* `  A )  e.  RR )
109107, 108sylan2 460 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol * `  A )  e.  RR )
110106, 109fsumrecl 12207 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol * `  A
)  e.  RR )
111 ovoliun.b . . . . . . 7  |-  ( ph  ->  B  e.  RR+ )
112111rpred 10390 . . . . . 6  |-  ( ph  ->  B  e.  RR )
113 2nn 9877 . . . . . . . 8  |-  2  e.  NN
114 nnnn0 9972 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  NN0 )
115 nnexpcl 11116 . . . . . . . 8  |-  ( ( 2  e.  NN  /\  n  e.  NN0 )  -> 
( 2 ^ n
)  e.  NN )
116113, 114, 115sylancr 644 . . . . . . 7  |-  ( n  e.  NN  ->  (
2 ^ n )  e.  NN )
117107, 116syl 15 . . . . . 6  |-  ( n  e.  ( 1 ... L )  ->  (
2 ^ n )  e.  NN )
118 nndivre 9781 . . . . . 6  |-  ( ( B  e.  RR  /\  ( 2 ^ n
)  e.  NN )  ->  ( B  / 
( 2 ^ n
) )  e.  RR )
119112, 117, 118syl2an 463 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( B  /  ( 2 ^ n ) )  e.  RR )
120106, 119fsumrecl 12207 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  e.  RR )
121110, 120readdcld 8862 . . 3  |-  ( ph  ->  ( sum_ n  e.  ( 1 ... L ) ( vol * `  A )  +  sum_ n  e.  ( 1 ... L ) ( B  /  ( 2 ^ n ) ) )  e.  RR )
122 ovoliun.r . . . 4  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
123122, 112readdcld 8862 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  +  B )  e.  RR )
124 relxp 4794 . . . . . . . . . . . . . . 15  |-  Rel  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )
125 relres 4983 . . . . . . . . . . . . . . 15  |-  Rel  (
( J " (
1 ... K ) )  |`  { n } )
126 opelxp 4719 . . . . . . . . . . . . . . . 16  |-  ( <.
x ,  y >.  e.  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  <-> 
( x  e.  {
n }  /\  y  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )
127 vex 2791 . . . . . . . . . . . . . . . . . 18  |-  y  e. 
_V
128127opelres 4960 . . . . . . . . . . . . . . . . 17  |-  ( <.
x ,  y >.  e.  ( ( J "
( 1 ... K
) )  |`  { n } )  <->  ( <. x ,  y >.  e.  ( J " ( 1 ... K ) )  /\  x  e.  {
n } ) )
129 ancom 437 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  { n }  /\  <. x ,  y
>.  e.  ( J "
( 1 ... K
) ) )  <->  ( <. x ,  y >.  e.  ( J " ( 1 ... K ) )  /\  x  e.  {
n } ) )
130 elsni 3664 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  { n }  ->  x  =  n )
131130opeq1d 3802 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  { n }  -> 
<. x ,  y >.  =  <. n ,  y
>. )
132131eleq1d 2349 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  { n }  ->  ( <. x ,  y
>.  e.  ( J "
( 1 ... K
) )  <->  <. n ,  y >.  e.  ( J " ( 1 ... K ) ) ) )
133 vex 2791 . . . . . . . . . . . . . . . . . . . 20  |-  n  e. 
_V
134133, 127elimasn 5038 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( ( J
" ( 1 ... K ) ) " { n } )  <->  <. n ,  y >.  e.  ( J " (
1 ... K ) ) )
135132, 134syl6bbr 254 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  { n }  ->  ( <. x ,  y
>.  e.  ( J "
( 1 ... K
) )  <->  y  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )
136135pm5.32i 618 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  { n }  /\  <. x ,  y
>.  e.  ( J "
( 1 ... K
) ) )  <->  ( x  e.  { n }  /\  y  e.  ( ( J " ( 1 ... K ) ) " { n } ) ) )
137128, 129, 1363bitr2ri 265 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  { n }  /\  y  e.  ( ( J " (
1 ... K ) )
" { n }
) )  <->  <. x ,  y >.  e.  (
( J " (
1 ... K ) )  |`  { n } ) )
138126, 137bitri 240 . . . . . . . . . . . . . . 15  |-  ( <.
x ,  y >.  e.  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  <->  <. x ,  y >.  e.  ( ( J "
( 1 ... K
) )  |`  { n } ) )
139124, 125, 138eqrelriiv 4781 . . . . . . . . . . . . . 14  |-  ( { n }  X.  (
( J " (
1 ... K ) )
" { n }
) )  =  ( ( J " (
1 ... K ) )  |`  { n } )
140 df-res 4701 . . . . . . . . . . . . . 14  |-  ( ( J " ( 1 ... K ) )  |`  { n } )  =  ( ( J
" ( 1 ... K ) )  i^i  ( { n }  X.  _V ) )
141139, 140eqtri 2303 . . . . . . . . . . . . 13  |-  ( { n }  X.  (
( J " (
1 ... K ) )
" { n }
) )  =  ( ( J " (
1 ... K ) )  i^i  ( { n }  X.  _V ) )
142141a1i 10 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  =  ( ( J
" ( 1 ... K ) )  i^i  ( { n }  X.  _V ) ) )
143142iuneq2dv 3926 . . . . . . . . . . 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 ) ) )
144 iunin2 3966 . . . . . . . . . . 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 ) )
145143, 144syl6eq 2331 . . . . . . . . . 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 ) ) )
146 relxp 4794 . . . . . . . . . . . . . 14  |-  Rel  ( NN  X.  NN )
147 relss 4775 . . . . . . . . . . . . . 14  |-  ( ( J " ( 1 ... K ) ) 
C_  ( NN  X.  NN )  ->  ( Rel  ( NN  X.  NN )  ->  Rel  ( J " ( 1 ... K
) ) ) )
14826, 146, 147ee10 1366 . . . . . . . . . . . . 13  |-  ( ph  ->  Rel  ( J "
( 1 ... K
) ) )
149 ovoliun.l2 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  A. w  e.  ( 1 ... K ) ( 1st `  ( J `  w )
)  <_  L )
150 ffn 5389 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J : NN --> ( NN 
X.  NN )  ->  J  Fn  NN )
15123, 150syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  J  Fn  NN )
152 fveq2 5525 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( j  =  ( J `  w )  ->  ( 1st `  j )  =  ( 1st `  ( J `  w )
) )
153152breq1d 4033 . . . . . . . . . . . . . . . . . . . . 21  |-  ( j  =  ( J `  w )  ->  (
( 1st `  j
)  <_  L  <->  ( 1st `  ( J `  w
) )  <_  L
) )
154153ralima 5758 . . . . . . . . . . . . . . . . . . . 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 )
)
155151, 13, 154sylancl 643 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( A. j  e.  ( J " (
1 ... K ) ) ( 1st `  j
)  <_  L  <->  A. w  e.  ( 1 ... K
) ( 1st `  ( J `  w )
)  <_  L )
)
156149, 155mpbird 223 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A. j  e.  ( J " ( 1 ... K ) ) ( 1st `  j
)  <_  L )
157156r19.21bi 2641 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  <_  L
)
15829, 82syl6eleq 2373 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  e.  (
ZZ>= `  1 ) )
159 ovoliun.l1 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  L  e.  ZZ )
160159adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  L  e.  ZZ )
161 elfz5 10790 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  j
)  e.  ( ZZ>= ` 
1 )  /\  L  e.  ZZ )  ->  (
( 1st `  j
)  e.  ( 1 ... L )  <->  ( 1st `  j )  <_  L
) )
162158, 160, 161syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( 1st `  j )  e.  ( 1 ... L
)  <->  ( 1st `  j
)  <_  L )
)
163157, 162mpbird 223 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  e.  ( 1 ... L ) )
164163ralrimiva 2626 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. j  e.  ( J " ( 1 ... K ) ) ( 1st `  j
)  e.  ( 1 ... L ) )
165 vex 2791 . . . . . . . . . . . . . . . . . 18  |-  x  e. 
_V
166165, 127op1std 6130 . . . . . . . . . . . . . . . . 17  |-  ( j  =  <. x ,  y
>.  ->  ( 1st `  j
)  =  x )
167166eleq1d 2349 . . . . . . . . . . . . . . . 16  |-  ( j  =  <. x ,  y
>.  ->  ( ( 1st `  j )  e.  ( 1 ... L )  <-> 
x  e.  ( 1 ... L ) ) )
168167rspccv 2881 . . . . . . . . . . . . . . 15  |-  ( A. j  e.  ( J " ( 1 ... K
) ) ( 1st `  j )  e.  ( 1 ... L )  ->  ( <. x ,  y >.  e.  ( J " ( 1 ... K ) )  ->  x  e.  ( 1 ... L ) ) )
169164, 168syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( J "
( 1 ... K
) )  ->  x  e.  ( 1 ... L
) ) )
170 opelxp 4719 . . . . . . . . . . . . . . 15  |-  ( <.
x ,  y >.  e.  ( U_ n  e.  ( 1 ... L
) { n }  X.  _V )  <->  ( x  e.  U_ n  e.  ( 1 ... L ) { n }  /\  y  e.  _V )
)
171127biantru 491 . . . . . . . . . . . . . . 15  |-  ( x  e.  U_ n  e.  ( 1 ... L
) { n }  <->  ( x  e.  U_ n  e.  ( 1 ... L
) { n }  /\  y  e.  _V ) )
172 iunid 3957 . . . . . . . . . . . . . . . 16  |-  U_ n  e.  ( 1 ... L
) { n }  =  ( 1 ... L )
173172eleq2i 2347 . . . . . . . . . . . . . . 15  |-  ( x  e.  U_ n  e.  ( 1 ... L
) { n }  <->  x  e.  ( 1 ... L ) )
174170, 171, 1733bitr2i 264 . . . . . . . . . . . . . 14  |-  ( <.
x ,  y >.  e.  ( U_ n  e.  ( 1 ... L
) { n }  X.  _V )  <->  x  e.  ( 1 ... L
) )
175169, 174syl6ibr 218 . . . . . . . . . . . . 13  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( J "
( 1 ... K
) )  ->  <. x ,  y >.  e.  (
U_ n  e.  ( 1 ... L ) { n }  X.  _V ) ) )
176148, 175relssdv 4779 . . . . . . . . . . . 12  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  ( U_ n  e.  ( 1 ... L
) { n }  X.  _V ) )
177 xpiundir 4744 . . . . . . . . . . . 12  |-  ( U_ n  e.  ( 1 ... L ) { n }  X.  _V )  =  U_ n  e.  ( 1 ... L
) ( { n }  X.  _V )
178176, 177syl6sseq 3224 . . . . . . . . . . 11  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  U_ n  e.  ( 1 ... L ) ( { n }  X.  _V ) )
179 df-ss 3166 . . . . . . . . . . 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 ) ) )
180178, 179sylib 188 . . . . . . . . . 10  |-  ( ph  ->  ( ( J "
( 1 ... K
) )  i^i  U_ n  e.  ( 1 ... L ) ( { n }  X.  _V ) )  =  ( J " ( 1 ... K ) ) )
181145, 180eqtrd 2315 . . . . . . . . 9  |-  ( ph  ->  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  =  ( J "
( 1 ... K
) ) )
182181, 104eqeltrd 2357 . . . . . . . 8  |-  ( ph  ->  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  e.  Fin )
183 ssiun2 3945 . . . . . . . 8  |-  ( n  e.  ( 1 ... L )  ->  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) 
C_  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )
184 ssfi 7083 . . . . . . . 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 )
185182, 183, 184syl2an 463 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  e.  Fin )
186 2ndconst 6208 . . . . . . . . . 10  |-  ( n  e.  _V  ->  ( 2nd  |`  ( { n }  X.  ( ( J
" ( 1 ... K ) ) " { n } ) ) ) : ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) -1-1-onto-> ( ( J " (
1 ... K ) )
" { n }
) )
187133, 186ax-mp 8 . . . . . . . . 9  |-  ( 2nd  |`  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) ) : ( { n }  X.  (
( J " (
1 ... K ) )
" { n }
) ) -1-1-onto-> ( ( J "
( 1 ... K
) ) " {
n } )
188 f1oeng 6880 . . . . . . . . 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 } ) )
189185, 187, 188sylancl 643 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) 
~~  ( ( J
" ( 1 ... K ) ) " { n } ) )
190 ensym 6910 . . . . . . . 8  |-  ( ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) 
~~  ( ( J
" ( 1 ... K ) ) " { n } )  ->  ( ( J
" ( 1 ... K ) ) " { n } ) 
~~  ( { n }  X.  ( ( J
" ( 1 ... K ) ) " { n } ) ) )
191189, 190syl 15 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( J " (
1 ... K ) )
" { n }
)  ~~  ( {
n }  X.  (
( J " (
1 ... K ) )
" { n }
) ) )
192 enfii 7080 . . . . . . 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 )
193185, 191, 192syl2anc 642 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( J " (
1 ... K ) )
" { n }
)  e.  Fin )
194 ffvelrn 5663 . . . . . . . . . . . . . 14  |-  ( ( F : NN --> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN )  /\  n  e.  NN )  ->  ( F `  n )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
19519, 107, 194syl2an 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( F `  n )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
19634, 35elmap 6796 . . . . . . . . . . . . 13  |-  ( ( F `  n )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) 
<->  ( F `  n
) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
197195, 196sylib 188 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( F `  n ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
198197adantrr 697 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( F `  n ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
199 imassrn 5025 . . . . . . . . . . . . . 14  |-  ( ( J " ( 1 ... K ) )
" { n }
)  C_  ran  ( J
" ( 1 ... K ) )
20026adantr 451 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( J " ( 1 ... K ) )  C_  ( NN  X.  NN ) )
201 rnss 4907 . . . . . . . . . . . . . . . 16  |-  ( ( J " ( 1 ... K ) ) 
C_  ( NN  X.  NN )  ->  ran  ( J " ( 1 ... K ) )  C_  ran  ( NN  X.  NN ) )
202200, 201syl 15 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  ( J " ( 1 ... K ) ) 
C_  ran  ( NN  X.  NN ) )
203 rnxpid 5109 . . . . . . . . . . . . . . 15  |-  ran  ( NN  X.  NN )  =  NN
204202, 203syl6sseq 3224 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  ( J " ( 1 ... K ) ) 
C_  NN )
205199, 204syl5ss 3190 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( J " (
1 ... K ) )
" { n }
)  C_  NN )
206205sseld 3179 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
i  e.  ( ( J " ( 1 ... K ) )
" { n }
)  ->  i  e.  NN ) )
207206impr 602 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  i  e.  NN )
208 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( ( F `  n
) : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  i  e.  NN )  ->  (
( F `  n
) `  i )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
209198, 207, 208syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( F `  n ) `  i )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
21018, 209sseldi 3178 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( F `  n ) `  i )  e.  ( RR  X.  RR ) )
211 xp2nd 6150 . . . . . . . . 9  |-  ( ( ( F `  n
) `  i )  e.  ( RR  X.  RR )  ->  ( 2nd `  (
( F `  n
) `  i )
)  e.  RR )
212210, 211syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( 2nd `  ( ( F `  n ) `  i
) )  e.  RR )
213 xp1st 6149 . . . . . . . . 9  |-  ( ( ( F `  n
) `  i )  e.  ( RR  X.  RR )  ->  ( 1st `  (
( F `  n
) `  i )
)  e.  RR )
214210, 213syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( 1st `  ( ( F `  n ) `  i
) )  e.  RR )
215212, 214resubcld 9211 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( 2nd `  ( ( F `
 n ) `  i ) )  -  ( 1st `  ( ( F `  n ) `
 i ) ) )  e.  RR )
216215anassrs 629 . . . . . 6  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) )  ->  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  e.  RR )
217193, 216fsumrecl 12207 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  e.  RR )
218112, 116, 118syl2an 463 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( B  /  ( 2 ^ n ) )  e.  RR )
219108, 218readdcld 8862 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( vol * `  A
)  +  ( B  /  ( 2 ^ n ) ) )  e.  RR )
220107, 219sylan2 460 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( vol * `  A )  +  ( B  /  ( 2 ^ n ) ) )  e.  RR )
221 eqid 2283 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  o.  ( F `  n
) )  =  ( ( abs  o.  -  )  o.  ( F `  n ) )
222 ovoliun.s . . . . . . . . . . . 12  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  ( F `  n ) ) )
223221, 222ovolsf 18832 . . . . . . . . . . 11  |-  ( ( F `  n ) : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  S : NN --> ( 0 [,) 
+oo ) )
224197, 223syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  S : NN --> ( 0 [,) 
+oo ) )
225 frn 5395 . . . . . . . . . 10  |-  ( S : NN --> ( 0 [,)  +oo )  ->  ran  S 
C_  ( 0 [,) 
+oo ) )
226224, 225syl 15 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S 
C_  ( 0 [,) 
+oo ) )
227 icossxr 10734 . . . . . . . . 9  |-  ( 0 [,)  +oo )  C_  RR*
228226, 227syl6ss 3191 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S 
C_  RR* )
229 supxrcl 10633 . . . . . . . 8  |-  ( ran 
S  C_  RR*  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
230228, 229syl 15 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
231 mnfxr 10456 . . . . . . . . 9  |-  -oo  e.  RR*
232231a1i 10 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  -oo  e.  RR* )
233109rexrd 8881 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol * `  A )  e.  RR* )
234 mnflt 10464 . . . . . . . . 9  |-  ( ( vol * `  A
)  e.  RR  ->  -oo 
<  ( vol * `  A ) )
235109, 234syl 15 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  -oo  <  ( vol * `  A
) )
236 ovoliun.x1 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  U.
ran  ( (,)  o.  ( F `  n ) ) )
237107, 236sylan2 460 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  A  C_ 
U. ran  ( (,)  o.  ( F `  n
) ) )
238222ovollb 18838 . . . . . . . . 9  |-  ( ( ( F `  n
) : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  ( F `  n
) ) )  -> 
( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  ) )
239197, 237, 238syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  )
)
240232, 233, 230, 235, 239xrltletrd 10492 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  -oo  <  sup ( ran  S ,  RR* ,  <  ) )
241 ovoliun.x2 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN )  ->  sup ( ran  S ,  RR* ,  <  )  <_  ( ( vol
* `  A )  +  ( B  / 
( 2 ^ n
) ) ) )
242107, 241sylan2 460 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  <_  (
( vol * `  A )  +  ( B  /  ( 2 ^ n ) ) ) )
243 xrre 10498 . . . . . . 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 )
244230, 220, 240, 242, 243syl22anc 1183 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR )
245 1z 10053 . . . . . . . . 9  |-  1  e.  ZZ
246245a1i 10 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  1  e.  ZZ )
247221ovolfsval 18830 . . . . . . . . 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 ) ) ) )
248197, 247sylan 457 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( ( abs  o.  -  )  o.  ( F `  n )
) `  i )  =  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) ) )
249221ovolfsf 18831 . . . . . . . . . . . . 13  |-  ( ( F `  n ) : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  ( F `  n
) ) : NN --> ( 0 [,)  +oo ) )
250197, 249syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( abs  o.  -  )  o.  ( F `  n
) ) : NN --> ( 0 [,)  +oo ) )
251 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( ( ( abs  o.  -  )  o.  ( F `  n )
) : NN --> ( 0 [,)  +oo )  /\  i  e.  NN )  ->  (
( ( abs  o.  -  )  o.  ( F `  n )
) `  i )  e.  ( 0 [,)  +oo ) )
252250, 251sylan 457 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( ( abs  o.  -  )  o.  ( F `  n )
) `  i )  e.  ( 0 [,)  +oo ) )
253248, 252eqeltrrd 2358 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  e.  ( 0 [,)  +oo ) )
254 elrege0 10746 . . . . . . . . . 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
) ) ) ) )
255253, 254sylib 188 . . . . . . . . 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
) ) ) ) )
256255simpld 445 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  e.  RR )
257255simprd 449 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  0  <_  ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) ) )
258 supxrub 10643 . . . . . . . . . . . . . . 15  |-  ( ( ran  S  C_  RR*  /\  z  e.  ran  S )  -> 
z  <_  sup ( ran  S ,  RR* ,  <  ) )
259228, 258sylan 457 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  z  e.  ran  S )  -> 
z  <_  sup ( ran  S ,  RR* ,  <  ) )
260259ralrimiva 2626 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  A. z  e.  ran  S  z  <_  sup ( ran  S ,  RR* ,  <  ) )
261 breq2 4027 . . . . . . . . . . . . . . 15  |-  ( x  =  sup ( ran 
S ,  RR* ,  <  )  ->  ( z  <_  x 
<->  z  <_  sup ( ran  S ,  RR* ,  <  ) ) )
262261ralbidv 2563 . . . . . . . . . . . . . 14  |-  ( x  =  sup ( ran 
S ,  RR* ,  <  )  ->  ( A. z  e.  ran  S  z  <_  x 
<-> 
A. z  e.  ran  S  z  <_  sup ( ran  S ,  RR* ,  <  ) ) )
263262rspcev 2884 . . . . . . . . . . . . 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 )
264244, 260, 263syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  E. x  e.  RR  A. z  e. 
ran  S  z  <_  x )
265 ffn 5389 . . . . . . . . . . . . . . 15  |-  ( S : NN --> ( 0 [,)  +oo )  ->  S  Fn  NN )
266224, 265syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  S  Fn  NN )
267 breq1 4026 . . . . . . . . . . . . . . 15  |-  ( z  =  ( S `  k )  ->  (
z  <_  x  <->  ( S `  k )  <_  x
) )
268267ralrn 5668 . . . . . . . . . . . . . 14  |-  ( S  Fn  NN  ->  ( A. z  e.  ran  S  z  <_  x  <->  A. k  e.  NN  ( S `  k )  <_  x
) )
269266, 268syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( A. z  e.  ran  S  z  <_  x  <->  A. k  e.  NN  ( S `  k )  <_  x
) )
270269rexbidv 2564 . . . . . . . . . . . 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
) )
271264, 270mpbid 201 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  E. x  e.  RR  A. k  e.  NN  ( S `  k )  <_  x
)
27282, 222, 246, 248, 256, 257, 271isumsup2 12305 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  S  ~~>  sup ( ran  S ,  RR ,  <  ) )
273222, 272syl5eqbrr 4057 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  seq  1 (  +  , 
( ( abs  o.  -  )  o.  ( F `  n )
) )  ~~>  sup ( ran  S ,  RR ,  <  ) )
274 climrel 11966 . . . . . . . . . 10  |-  Rel  ~~>
275274releldmi 4915 . . . . . . . . 9  |-  (  seq  1 (  +  , 
( ( abs  o.  -  )  o.  ( F `  n )
) )  ~~>  sup ( ran  S ,  RR ,  <  )  ->  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  ( F `  n ) ) )  e.  dom  ~~>  )
276273, 275syl 15 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  seq  1 (  +  , 
( ( abs  o.  -  )  o.  ( F `  n )
) )  e.  dom  ~~>  )
27782, 246, 193, 205, 248, 256, 257, 276isumless 12304 . . . . . . 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
) ) ) )
27882, 222, 246, 248, 256, 257, 271isumsup 12306 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  NN  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  =  sup ( ran 
S ,  RR ,  <  ) )
279 0re 8838 . . . . . . . . . . 11  |-  0  e.  RR
280 pnfxr 10455 . . . . . . . . . . 11  |-  +oo  e.  RR*
281 icossre 10730 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
282279, 280, 281mp2an 653 . . . . . . . . . 10  |-  ( 0 [,)  +oo )  C_  RR
283226, 282syl6ss 3191 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S 
C_  RR )
284 1nn 9757 . . . . . . . . . . . 12  |-  1  e.  NN
285 fdm 5393 . . . . . . . . . . . . 13  |-  ( S : NN --> ( 0 [,)  +oo )  ->  dom  S  =  NN )
286224, 285syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  dom  S  =  NN )
287284, 286syl5eleqr 2370 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  1  e.  dom  S )
288 ne0i 3461 . . . . . . . . . . 11  |-  ( 1  e.  dom  S  ->  dom  S  =/=  (/) )
289287, 288syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  dom  S  =/=  (/) )
290 dm0rn0 4895 . . . . . . . . . . 11  |-  ( dom 
S  =  (/)  <->  ran  S  =  (/) )
291290necon3bii 2478 . . . . . . . . . 10  |-  ( dom 
S  =/=  (/)  <->  ran  S  =/=  (/) )
292289, 291sylib 188 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S  =/=  (/) )
293 supxrre 10646 . . . . . . . . 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 ,  <  ) )
294283, 292, 264, 293syl3anc 1182 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  S ,  RR ,  <  ) )
295278, 294eqtr4d 2318 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  NN  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  =  sup ( ran 
S ,  RR* ,  <  ) )
296277, 295breqtrd 4047 . . . . . 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* ,  <  ) )
297217, 244, 220, 296, 242letrd 8973 . . . . 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 ) ) ) )
298106, 217, 220, 297fsumle 12257 . . . 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 ) ) ) )
299 vex 2791 . . . . . . . . . . 11  |-  i  e. 
_V
300133, 299op1std 6130 . . . . . . . . . 10  |-  ( j  =  <. n ,  i
>.  ->  ( 1st `  j
)  =  n )
301300fveq2d 5529 . . . . . . . . 9  |-  ( j  =  <. n ,  i
>.  ->  ( F `  ( 1st `  j ) )  =  ( F `
 n ) )
302133, 299op2ndd 6131 . . . . . . . . 9  |-  ( j  =  <. n ,  i
>.  ->  ( 2nd `  j
)  =  i )
303301, 302fveq12d 5531 . . . . . . . 8  |-  ( j  =  <. n ,  i
>.  ->  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) )  =  ( ( F `
 n ) `  i ) )
304303fveq2d 5529 . . . . . . 7  |-  ( j  =  <. n ,  i
>.  ->  ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  =  ( 2nd `  (
( F `  n
) `  i )
) )
305303fveq2d 5529 . . . . . . 7  |-  ( j  =  <. n ,  i
>.  ->  ( 1st `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  =  ( 1st `  (
( F `  n
) `  i )
) )
306304, 305oveq12d 5876 . . . . . 6  |-  ( j  =  <. n ,  i
>.  ->  ( ( 2nd `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) )  -  ( 1st `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) ) )  =  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) ) )
307215recnd 8861 . . . . . 6  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( 2nd `  ( ( F `
 n ) `  i ) )  -  ( 1st `  ( ( F `  n ) `
 i ) ) )  e.  CC )
308306, 106, 193, 307fsum2d 12234 . . . . 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 ) ) ) ) )
309181sumeq1d 12174 . . . . 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 ) ) ) ) )
310308, 309eqtrd 2315 . . . 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 ) ) ) ) )
311109recnd 8861 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol * `  A )  e.  CC )
312119recnd 8861 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( B  /  ( 2 ^ n ) )  e.  CC )
313106, 311, 312fsumadd 12211 . . . 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 ) ) ) )
314298, 310, 3133brtr3d 4052 . . 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 ) ) ) )
315245a1i 10 . . . . . . . . 9  |-  ( ph  ->  1  e.  ZZ )
316 simpr 447 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  NN )
317 ovoliun.g . . . . . . . . . . . 12  |-  G  =  ( n  e.  NN  |->  ( vol * `  A
) )
318317fvmpt2 5608 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  ( vol * `  A
)  e.  RR )  ->  ( G `  n )  =  ( vol * `  A
) )
319316, 108, 318syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  =  ( vol * `  A ) )
320319, 108eqeltrd 2357 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
32182, 315, 320serfre 11075 . . . . . . . 8  |-  ( ph  ->  seq  1 (  +  ,  G ) : NN --> RR )
322 ovoliun.t . . . . . . . . 9  |-  T  =  seq  1 (  +  ,  G )
323322feq1i 5383 . . . . . . . 8  |-  ( T : NN --> RR  <->  seq  1
(  +  ,  G
) : NN --> RR )
324321, 323sylibr 203 . . . . . . 7  |-  ( ph  ->  T : NN --> RR )
325 frn 5395 . . . . . . 7  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
326324, 325syl 15 . . . . . 6  |-  ( ph  ->  ran  T  C_  RR )
327 ressxr 8876 . . . . . 6  |-  RR  C_  RR*
328326, 327syl6ss 3191 . . . . 5  |-  ( ph  ->  ran  T  C_  RR* )
329107, 319sylan2 460 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( G `  n )  =  ( vol * `  A ) )
330 1re 8837 . . . . . . . . . . 11  |-  1  e.  RR
331330a1i 10 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
332 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( J : NN --> ( NN 
X.  NN )  /\  1  e.  NN )  ->  ( J `  1
)  e.  ( NN 
X.  NN ) )
33323, 284, 332sylancl 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( J `  1
)  e.  ( NN 
X.  NN ) )
334 xp1st 6149 . . . . . . . . . . . 12  |-  ( ( J `  1 )  e.  ( NN  X.  NN )  ->  ( 1st `  ( J `  1
) )  e.  NN )
335333, 334syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  e.  NN )
336335nnred 9761 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  e.  RR )
337159zred 10117 . . . . . . . . . 10  |-  ( ph  ->  L  e.  RR )
338335nnge1d 9788 . . . . . . . . . 10  |-  ( ph  ->  1  <_  ( 1st `  ( J `  1
) ) )
339 eluzfz1 10803 . . . . . . . . . . . 12  |-  ( K  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... K
) )
34083, 339syl 15 . . . . . . . . . . 11  |-  ( ph  ->  1  e.  ( 1 ... K ) )
341 fveq2 5525 . . . . . . . . . . . . . 14  |-  ( w  =  1  ->  ( J `  w )  =  ( J ` 
1 ) )
342341fveq2d 5529 . . . . . . . . . . . . 13  |-  ( w  =  1  ->  ( 1st `  ( J `  w ) )  =  ( 1st `  ( J `  1 )
) )
343342breq1d 4033 . . . . . . . . . . . 12  |-  ( w  =  1  ->  (
( 1st `  ( J `  w )
)  <_  L  <->  ( 1st `  ( J `  1
) )  <_  L
) )
344343rspcv 2880 . . . . . . . . . . 11  |-  ( 1  e.  ( 1 ... K )  ->  ( A. w  e.  (
1 ... K ) ( 1st `  ( J `
 w ) )  <_  L  ->  ( 1st `  ( J ` 
1 ) )  <_  L ) )
345340, 149, 344sylc 56 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  <_  L )
346331, 336, 337, 338, 345letrd 8973 . . . . . . . . 9  |-  ( ph  ->  1  <_  L )
347 elnnz1 10049 . . . . . . . . 9  |-  ( L  e.  NN  <->  ( L  e.  ZZ  /\  1  <_  L ) )
348159, 346, 347sylanbrc 645 . . . . . . . 8  |-  ( ph  ->  L  e.  NN )
349348, 82syl6eleq 2373 . . . . . . 7  |-  ( ph  ->  L  e.  ( ZZ>= ` 
1 ) )
350329, 349, 311fsumser 12203 . . . . . 6  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol * `  A
)  =  (  seq  1 (  +  ,  G ) `  L
) )
351 seqfn 11058 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  seq  1 (  +  ,  G )  Fn  ( ZZ>=
`  1 ) )
352315, 351syl 15 . . . . . . . 8  |-  ( ph  ->  seq  1 (  +  ,  G )  Fn  ( ZZ>= `  1 )
)
353 fnfvelrn 5662 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  G )  Fn  ( ZZ>= `  1 )  /\  L  e.  ( ZZ>=
`  1 ) )  ->  (  seq  1
(  +  ,  G
) `  L )  e.  ran  seq  1 (  +  ,  G ) )
354352, 349, 353syl2anc 642 . . . . . . 7  |-  ( ph  ->  (  seq  1 (  +  ,  G ) `
 L )  e. 
ran  seq  1 (  +  ,  G ) )
355322rneqi 4905 . . . . . . 7  |-  ran  T  =  ran  seq  1 (  +  ,  G )
356354, 355syl6eleqr 2374 . . . . . 6  |-  ( ph  ->  (  seq  1 (  +  ,  G ) `
 L )  e. 
ran  T )
357350, 356eqeltrd 2357 . . . . 5  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol * `  A
)  e.  ran  T
)
358 supxrub 10643 . . . . 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* ,  <  ) )
359328, 357, 358syl2anc 642 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol * `  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
360112recnd 8861 . . . . . 6  |-  ( ph  ->  B  e.  CC )
361 geo2sum 12329 . . . . . 6  |-  ( ( L  e.  NN  /\  B  e.  CC )  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  =  ( B  -  ( B  /  (
2 ^ L ) ) ) )
362348, 360, 361syl2anc 642 . . . . 5  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  =  ( B  -  ( B  /  (
2 ^ L ) ) ) )
363348nnnn0d 10018 . . . . . . . . . 10  |-  ( ph  ->  L  e.  NN0 )
364 nnexpcl 11116 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  L  e.  NN0 )  -> 
( 2 ^ L
)  e.  NN )
365113, 363, 364sylancr 644 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ L
)  e.  NN )
366365nnrpd 10389 . . . . . . . 8  |-  ( ph  ->  ( 2 ^ L
)  e.  RR+ )
367111, 366rpdivcld 10407 . . . . . . 7  |-  ( ph  ->  ( B  /  (
2 ^ L ) )  e.  RR+ )
368367rpge0d 10394 . . . . . 6  |-  ( ph  ->  0  <_  ( B  /  ( 2 ^ L ) ) )
369112, 365nndivred 9794 . . . . . . 7  |-  ( ph  ->  ( B  /  (
2 ^ L ) )  e.  RR )
370112, 369subge02d 9364 . . . . . 6  |-  ( ph  ->  ( 0  <_  ( B  /  ( 2 ^ L ) )  <->  ( B  -  ( B  / 
( 2 ^ L
) ) )  <_  B ) )
371368, 370mpbid 201 . . . . 5  |-  ( ph  ->  ( B  -  ( B  /  ( 2 ^ L ) ) )  <_  B )
372362, 371eqbrtrd 4043 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  <_  B )
373110, 120, 122, 112, 359, 372le2addd 9390 . . 3  |-  ( ph  ->  ( sum_ n  e.  ( 1 ... L ) ( vol * `  A )  +  sum_ n  e.  ( 1 ... L ) ( B  /  ( 2 ^ n ) ) )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  B ) )
374105, 121, 123, 314, 373letrd 8973 . 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 ) )
37598, 374eqbrtrrd 4045 1  |-  ( ph  ->  ( U `  K
)  <_  ( sup ( ran  T ,  RR* ,  <  )  +  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   U.cuni 3827   U_ciun 3905   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   Rel wrel 4694    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772    ~~ cen 6860   Fincfn 6863   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   (,)cioo 10656   [,)cico 10658   ...cfz 10782    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958   sum_csu 12158   vol *covol 18822
This theorem is referenced by:  ovoliunlem2  18862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ioo 10660  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-ovol 18824
  Copyright terms: Public domain W3C validator