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Theorem ovoliunlem1 22039
Description: Lemma for ovoliun 22042. (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 5872 . . . . . . . . 9  |-  ( j  =  ( J `  m )  ->  ( 1st `  j )  =  ( 1st `  ( J `  m )
) )
21fveq2d 5876 . . . . . . . 8  |-  ( j  =  ( J `  m )  ->  ( F `  ( 1st `  j ) )  =  ( F `  ( 1st `  ( J `  m ) ) ) )
3 fveq2 5872 . . . . . . . 8  |-  ( j  =  ( J `  m )  ->  ( 2nd `  j )  =  ( 2nd `  ( J `  m )
) )
42, 3fveq12d 5878 . . . . . . 7  |-  ( j  =  ( J `  m )  ->  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) )  =  ( ( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )
54fveq2d 5876 . . . . . 6  |-  ( j  =  ( J `  m )  ->  ( 2nd `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  =  ( 2nd `  ( ( F `  ( 1st `  ( J `
 m ) ) ) `  ( 2nd `  ( J `  m
) ) ) ) )
64fveq2d 5876 . . . . . 6  |-  ( j  =  ( J `  m )  ->  ( 1st `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  =  ( 1st `  ( ( F `  ( 1st `  ( J `
 m ) ) ) `  ( 2nd `  ( J `  m
) ) ) ) )
75, 6oveq12d 6314 . . . . 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 12086 . . . . 5  |-  ( ph  ->  ( 1 ... K
)  e.  Fin )
9 ovoliun.j . . . . . . 7  |-  ( ph  ->  J : NN -1-1-onto-> ( NN  X.  NN ) )
10 f1of1 5821 . . . . . . 7  |-  ( J : NN -1-1-onto-> ( NN  X.  NN )  ->  J : NN -1-1-> ( NN  X.  NN ) )
119, 10syl 16 . . . . . 6  |-  ( ph  ->  J : NN -1-1-> ( NN  X.  NN ) )
12 elfznn 11739 . . . . . . 7  |-  ( m  e.  ( 1 ... K )  ->  m  e.  NN )
1312ssriv 3503 . . . . . 6  |-  ( 1 ... K )  C_  NN
14 f1ores 5836 . . . . . 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 662 . . . . 5  |-  ( ph  ->  ( J  |`  (
1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " (
1 ... K ) ) )
16 fvres 5886 . . . . . 6  |-  ( m  e.  ( 1 ... K )  ->  (
( J  |`  (
1 ... K ) ) `
 m )  =  ( J `  m
) )
1716adantl 466 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( J  |`  (
1 ... K ) ) `
 m )  =  ( J `  m
) )
18 inss2 3715 . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  F : NN
--> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
21 imassrn 5358 . . . . . . . . . . . . . . 15  |-  ( J
" ( 1 ... K ) )  C_  ran  J
22 f1of 5822 . . . . . . . . . . . . . . . . 17  |-  ( J : NN -1-1-onto-> ( NN  X.  NN )  ->  J : NN --> ( NN  X.  NN ) )
239, 22syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  J : NN --> ( NN 
X.  NN ) )
24 frn 5743 . . . . . . . . . . . . . . . 16  |-  ( J : NN --> ( NN 
X.  NN )  ->  ran  J  C_  ( NN  X.  NN ) )
2523, 24syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  J  C_  ( NN  X.  NN ) )
2621, 25syl5ss 3510 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  ( NN  X.  NN ) )
2726sselda 3499 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  j  e.  ( NN  X.  NN ) )
28 xp1st 6829 . . . . . . . . . . . . 13  |-  ( j  e.  ( NN  X.  NN )  ->  ( 1st `  j )  e.  NN )
2927, 28syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  e.  NN )
3020, 29ffvelrnd 6033 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( F `  ( 1st `  j
) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
31 reex 9600 . . . . . . . . . . . . . 14  |-  RR  e.  _V
3231, 31xpex 6603 . . . . . . . . . . . . 13  |-  ( RR 
X.  RR )  e. 
_V
3332inex2 4598 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
34 nnex 10562 . . . . . . . . . . . 12  |-  NN  e.  _V
3533, 34elmap 7466 . . . . . . . . . . 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 6830 . . . . . . . . . . 11  |-  ( j  e.  ( NN  X.  NN )  ->  ( 2nd `  j )  e.  NN )
3827, 37syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 2nd `  j )  e.  NN )
3936, 38ffvelrnd 6033 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
4018, 39sseldi 3497 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) )  e.  ( RR 
X.  RR ) )
41 xp2nd 6830 . . . . . . . 8  |-  ( ( ( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) )  e.  ( RR  X.  RR )  ->  ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  e.  RR )
4240, 41syl 16 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 2nd `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) )  e.  RR )
43 xp1st 6829 . . . . . . . 8  |-  ( ( ( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) )  e.  ( RR  X.  RR )  ->  ( 1st `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  e.  RR )
4440, 43syl 16 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) )  e.  RR )
4542, 44resubcld 10008 . . . . . 6  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( 2nd `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  e.  RR )
4645recnd 9639 . . . . 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 13557 . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  F : NN
--> ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
4923ffvelrnda 6032 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  ( J `
 k )  e.  ( NN  X.  NN ) )
50 xp1st 6829 . . . . . . . . . . . 12  |-  ( ( J `  k )  e.  ( NN  X.  NN )  ->  ( 1st `  ( J `  k
) )  e.  NN )
5149, 50syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  ( 1st `  ( J `  k
) )  e.  NN )
5248, 51ffvelrnd 6033 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 ( 1st `  ( J `  k )
) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
5333, 34elmap 7466 . . . . . . . . . 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 6830 . . . . . . . . . 10  |-  ( ( J `  k )  e.  ( NN  X.  NN )  ->  ( 2nd `  ( J `  k
) )  e.  NN )
5649, 55syl 16 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( 2nd `  ( J `  k
) )  e.  NN )
5754, 56ffvelrnd 6033 . . . . . . . 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 6056 . . . . . . 7  |-  ( ph  ->  H : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
60 eqid 2457 . . . . . . . 8  |-  ( ( abs  o.  -  )  o.  H )  =  ( ( abs  o.  -  )  o.  H )
6160ovolfsval 22008 . . . . . . 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 477 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( ( abs  o.  -  )  o.  H
) `  m )  =  ( ( 2nd `  ( H `  m
) )  -  ( 1st `  ( H `  m ) ) ) )
6312adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  m  e.  NN )
64 fveq2 5872 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( J `  k )  =  ( J `  m ) )
6564fveq2d 5876 . . . . . . . . . . . 12  |-  ( k  =  m  ->  ( 1st `  ( J `  k ) )  =  ( 1st `  ( J `  m )
) )
6665fveq2d 5876 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( F `  ( 1st `  ( J `  k
) ) )  =  ( F `  ( 1st `  ( J `  m ) ) ) )
6764fveq2d 5876 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( 2nd `  ( J `  k ) )  =  ( 2nd `  ( J `  m )
) )
6866, 67fveq12d 5878 . . . . . . . . . 10  |-  ( k  =  m  ->  (
( F `  ( 1st `  ( J `  k ) ) ) `
 ( 2nd `  ( J `  k )
) )  =  ( ( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )
69 fvex 5882 . . . . . . . . . 10  |-  ( ( F `  ( 1st `  ( J `  m
) ) ) `  ( 2nd `  ( J `
 m ) ) )  e.  _V
7068, 58, 69fvmpt 5956 . . . . . . . . 9  |-  ( m  e.  NN  ->  ( H `  m )  =  ( ( F `
 ( 1st `  ( J `  m )
) ) `  ( 2nd `  ( J `  m ) ) ) )
7163, 70syl 16 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( H `  m )  =  ( ( F `
 ( 1st `  ( J `  m )
) ) `  ( 2nd `  ( J `  m ) ) ) )
7271fveq2d 5876 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 2nd `  ( H `  m ) )  =  ( 2nd `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) ) )
7371fveq2d 5876 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 1st `  ( H `  m ) )  =  ( 1st `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) ) )
7472, 73oveq12d 6314 . . . . . 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 2498 . . . . 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 11141 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
7876, 77syl6eleq 2555 . . . . 5  |-  ( ph  ->  K  e.  ( ZZ>= ` 
1 ) )
79 ffvelrn 6030 . . . . . . . . . . 11  |-  ( ( H : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  m  e.  NN )  ->  ( H `  m )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
8059, 12, 79syl2an 477 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( H `  m )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
8118, 80sseldi 3497 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( H `  m )  e.  ( RR  X.  RR ) )
82 xp2nd 6830 . . . . . . . . 9  |-  ( ( H `  m )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( H `  m
) )  e.  RR )
8381, 82syl 16 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 2nd `  ( H `  m ) )  e.  RR )
84 xp1st 6829 . . . . . . . . 9  |-  ( ( H `  m )  e.  ( RR  X.  RR )  ->  ( 1st `  ( H `  m
) )  e.  RR )
8581, 84syl 16 . . . . . . . 8  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 1st `  ( H `  m ) )  e.  RR )
8683, 85resubcld 10008 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( 2nd `  ( H `  m )
)  -  ( 1st `  ( H `  m
) ) )  e.  RR )
8786recnd 9639 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( 2nd `  ( H `  m )
)  -  ( 1st `  ( H `  m
) ) )  e.  CC )
8874, 87eqeltrrd 2546 . . . . 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 13564 . . . 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 2498 . . 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 5873 . . 3  |-  ( U `
 K )  =  (  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  H ) ) `  K )
9390, 92syl6eqr 2516 . 2  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  =  ( U `  K
) )
94 f1oeng 7553 . . . . . . 7  |-  ( ( ( 1 ... K
)  e.  Fin  /\  ( J  |`  ( 1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " (
1 ... K ) ) )  ->  ( 1 ... K )  ~~  ( J " ( 1 ... K ) ) )
958, 15, 94syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 1 ... K
)  ~~  ( J " ( 1 ... K
) ) )
9695ensymd 7585 . . . . 5  |-  ( ph  ->  ( J " (
1 ... K ) ) 
~~  ( 1 ... K ) )
97 enfii 7756 . . . . 5  |-  ( ( ( 1 ... K
)  e.  Fin  /\  ( J " ( 1 ... K ) ) 
~~  ( 1 ... K ) )  -> 
( J " (
1 ... K ) )  e.  Fin )
988, 96, 97syl2anc 661 . . . 4  |-  ( ph  ->  ( J " (
1 ... K ) )  e.  Fin )
9998, 45fsumrecl 13568 . . 3  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  e.  RR )
100 fzfid 12086 . . . . 5  |-  ( ph  ->  ( 1 ... L
)  e.  Fin )
101 elfznn 11739 . . . . . 6  |-  ( n  e.  ( 1 ... L )  ->  n  e.  NN )
102 ovoliun.v . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( vol* `  A )  e.  RR )
103101, 102sylan2 474 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol* `  A )  e.  RR )
104100, 103fsumrecl 13568 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  e.  RR )
105 ovoliun.b . . . . . . 7  |-  ( ph  ->  B  e.  RR+ )
106105rpred 11281 . . . . . 6  |-  ( ph  ->  B  e.  RR )
107 2nn 10714 . . . . . . . 8  |-  2  e.  NN
108 nnnn0 10823 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  NN0 )
109 nnexpcl 12182 . . . . . . . 8  |-  ( ( 2  e.  NN  /\  n  e.  NN0 )  -> 
( 2 ^ n
)  e.  NN )
110107, 108, 109sylancr 663 . . . . . . 7  |-  ( n  e.  NN  ->  (
2 ^ n )  e.  NN )
111101, 110syl 16 . . . . . 6  |-  ( n  e.  ( 1 ... L )  ->  (
2 ^ n )  e.  NN )
112 nndivre 10592 . . . . . 6  |-  ( ( B  e.  RR  /\  ( 2 ^ n
)  e.  NN )  ->  ( B  / 
( 2 ^ n
) )  e.  RR )
113106, 111, 112syl2an 477 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( B  /  ( 2 ^ n ) )  e.  RR )
114100, 113fsumrecl 13568 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  e.  RR )
115104, 114readdcld 9640 . . 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 9640 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  +  B )  e.  RR )
118 relxp 5119 . . . . . . . . . . . . . . 15  |-  Rel  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )
119 relres 5311 . . . . . . . . . . . . . . 15  |-  Rel  (
( J " (
1 ... K ) )  |`  { n } )
120 opelxp 5038 . . . . . . . . . . . . . . . 16  |-  ( <.
x ,  y >.  e.  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  <-> 
( x  e.  {
n }  /\  y  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )
121 vex 3112 . . . . . . . . . . . . . . . . . 18  |-  y  e. 
_V
122121opelres 5289 . . . . . . . . . . . . . . . . 17  |-  ( <.
x ,  y >.  e.  ( ( J "
( 1 ... K
) )  |`  { n } )  <->  ( <. x ,  y >.  e.  ( J " ( 1 ... K ) )  /\  x  e.  {
n } ) )
123 ancom 450 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  { n }  /\  <. x ,  y
>.  e.  ( J "
( 1 ... K
) ) )  <->  ( <. x ,  y >.  e.  ( J " ( 1 ... K ) )  /\  x  e.  {
n } ) )
124 elsni 4057 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  { n }  ->  x  =  n )
125124opeq1d 4225 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  { n }  -> 
<. x ,  y >.  =  <. n ,  y
>. )
126125eleq1d 2526 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  { n }  ->  ( <. x ,  y
>.  e.  ( J "
( 1 ... K
) )  <->  <. n ,  y >.  e.  ( J " ( 1 ... K ) ) ) )
127 vex 3112 . . . . . . . . . . . . . . . . . . . 20  |-  n  e. 
_V
128127, 121elimasn 5372 . . . . . . . . . . . . . . . . . . 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 637 . . . . . . . . . . . . . . . . 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 5106 . . . . . . . . . . . . . 14  |-  ( { n }  X.  (
( J " (
1 ... K ) )
" { n }
) )  =  ( ( J " (
1 ... K ) )  |`  { n } )
134 df-res 5020 . . . . . . . . . . . . . 14  |-  ( ( J " ( 1 ... K ) )  |`  { n } )  =  ( ( J
" ( 1 ... K ) )  i^i  ( { n }  X.  _V ) )
135133, 134eqtri 2486 . . . . . . . . . . . . 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 4354 . . . . . . . . . . 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 4396 . . . . . . . . . . 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 2514 . . . . . . . . . 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 5119 . . . . . . . . . . . . . 14  |-  Rel  ( NN  X.  NN )
141 relss 5099 . . . . . . . . . . . . . 14  |-  ( ( J " ( 1 ... K ) ) 
C_  ( NN  X.  NN )  ->  ( Rel  ( NN  X.  NN )  ->  Rel  ( J " ( 1 ... K
) ) ) )
14226, 140, 141mpisyl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  Rel  ( J "
( 1 ... K
) ) )
143 ovoliun.l2 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  A. w  e.  ( 1 ... K ) ( 1st `  ( J `  w )
)  <_  L )
144 ffn 5737 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J : NN --> ( NN 
X.  NN )  ->  J  Fn  NN )
14523, 144syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  J  Fn  NN )
146 fveq2 5872 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( j  =  ( J `  w )  ->  ( 1st `  j )  =  ( 1st `  ( J `  w )
) )
147146breq1d 4466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( j  =  ( J `  w )  ->  (
( 1st `  j
)  <_  L  <->  ( 1st `  ( J `  w
) )  <_  L
) )
148147ralima 6153 . . . . . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  <_  L
)
15229, 77syl6eleq 2555 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  e.  (
ZZ>= `  1 ) )
153 ovoliun.l1 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  L  e.  ZZ )
154153adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  L  e.  ZZ )
155 elfz5 11705 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  j
)  e.  ( ZZ>= ` 
1 )  /\  L  e.  ZZ )  ->  (
( 1st `  j
)  e.  ( 1 ... L )  <->  ( 1st `  j )  <_  L
) )
156152, 154, 155syl2anc 661 . . . . . . . . . . . . . . . . 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 2871 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. j  e.  ( J " ( 1 ... K ) ) ( 1st `  j
)  e.  ( 1 ... L ) )
159 vex 3112 . . . . . . . . . . . . . . . . . 18  |-  x  e. 
_V
160159, 121op1std 6809 . . . . . . . . . . . . . . . . 17  |-  ( j  =  <. x ,  y
>.  ->  ( 1st `  j
)  =  x )
161160eleq1d 2526 . . . . . . . . . . . . . . . 16  |-  ( j  =  <. x ,  y
>.  ->  ( ( 1st `  j )  e.  ( 1 ... L )  <-> 
x  e.  ( 1 ... L ) ) )
162161rspccv 3207 . . . . . . . . . . . . . . 15  |-  ( A. j  e.  ( J " ( 1 ... K
) ) ( 1st `  j )  e.  ( 1 ... L )  ->  ( <. x ,  y >.  e.  ( J " ( 1 ... K ) )  ->  x  e.  ( 1 ... L ) ) )
163158, 162syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( J "
( 1 ... K
) )  ->  x  e.  ( 1 ... L
) ) )
164 opelxp 5038 . . . . . . . . . . . . . . 15  |-  ( <.
x ,  y >.  e.  ( U_ n  e.  ( 1 ... L
) { n }  X.  _V )  <->  ( x  e.  U_ n  e.  ( 1 ... L ) { n }  /\  y  e.  _V )
)
165121biantru 505 . . . . . . . . . . . . . . 15  |-  ( x  e.  U_ n  e.  ( 1 ... L
) { n }  <->  ( x  e.  U_ n  e.  ( 1 ... L
) { n }  /\  y  e.  _V ) )
166 iunid 4387 . . . . . . . . . . . . . . . 16  |-  U_ n  e.  ( 1 ... L
) { n }  =  ( 1 ... L )
167166eleq2i 2535 . . . . . . . . . . . . . . 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 5104 . . . . . . . . . . . 12  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  ( U_ n  e.  ( 1 ... L
) { n }  X.  _V ) )
171 xpiundir 5064 . . . . . . . . . . . 12  |-  ( U_ n  e.  ( 1 ... L ) { n }  X.  _V )  =  U_ n  e.  ( 1 ... L
) ( { n }  X.  _V )
172170, 171syl6sseq 3545 . . . . . . . . . . 11  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  U_ n  e.  ( 1 ... L ) ( { n }  X.  _V ) )
173 df-ss 3485 . . . . . . . . . . 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 2498 . . . . . . . . 9  |-  ( ph  ->  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  =  ( J "
( 1 ... K
) ) )
176175, 98eqeltrd 2545 . . . . . . . 8  |-  ( ph  ->  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  e.  Fin )
177 ssiun2 4375 . . . . . . . 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 7759 . . . . . . . 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 477 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  e.  Fin )
180 2ndconst 6888 . . . . . . . . . 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 7553 . . . . . . . . 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 662 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) ) 
~~  ( ( J
" ( 1 ... K ) ) " { n } ) )
184183ensymd 7585 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( J " (
1 ... K ) )
" { n }
)  ~~  ( {
n }  X.  (
( J " (
1 ... K ) )
" { n }
) ) )
185 enfii 7756 . . . . . . 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 661 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( J " (
1 ... K ) )
" { n }
)  e.  Fin )
187 ffvelrn 6030 . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( F `  n )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
18933, 34elmap 7466 . . . . . . . . . . . . 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 716 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( F `  n ) : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
192 imassrn 5358 . . . . . . . . . . . . . 14  |-  ( ( J " ( 1 ... K ) )
" { n }
)  C_  ran  ( J
" ( 1 ... K ) )
19326adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( J " ( 1 ... K ) )  C_  ( NN  X.  NN ) )
194 rnss 5241 . . . . . . . . . . . . . . . 16  |-  ( ( J " ( 1 ... K ) ) 
C_  ( NN  X.  NN )  ->  ran  ( J " ( 1 ... K ) )  C_  ran  ( NN  X.  NN ) )
195193, 194syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  ( J " ( 1 ... K ) ) 
C_  ran  ( NN  X.  NN ) )
196 rnxpid 5447 . . . . . . . . . . . . . . 15  |-  ran  ( NN  X.  NN )  =  NN
197195, 196syl6sseq 3545 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  ( J " ( 1 ... K ) ) 
C_  NN )
198192, 197syl5ss 3510 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( J " (
1 ... K ) )
" { n }
)  C_  NN )
199198sseld 3498 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
i  e.  ( ( J " ( 1 ... K ) )
" { n }
)  ->  i  e.  NN ) )
200199impr 619 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  i  e.  NN )
201191, 200ffvelrnd 6033 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( F `  n ) `  i )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
20218, 201sseldi 3497 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( F `  n ) `  i )  e.  ( RR  X.  RR ) )
203 xp2nd 6830 . . . . . . . . 9  |-  ( ( ( F `  n
) `  i )  e.  ( RR  X.  RR )  ->  ( 2nd `  (
( F `  n
) `  i )
)  e.  RR )
204202, 203syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( 2nd `  ( ( F `  n ) `  i
) )  e.  RR )
205 xp1st 6829 . . . . . . . . 9  |-  ( ( ( F `  n
) `  i )  e.  ( RR  X.  RR )  ->  ( 1st `  (
( F `  n
) `  i )
)  e.  RR )
206202, 205syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( 1st `  ( ( F `  n ) `  i
) )  e.  RR )
207204, 206resubcld 10008 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( 2nd `  ( ( F `
 n ) `  i ) )  -  ( 1st `  ( ( F `  n ) `
 i ) ) )  e.  RR )
208207anassrs 648 . . . . . 6  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) )  ->  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  e.  RR )
209186, 208fsumrecl 13568 . . . . 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 477 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN )  ->  ( B  /  ( 2 ^ n ) )  e.  RR )
211102, 210readdcld 9640 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( vol* `  A
)  +  ( B  /  ( 2 ^ n ) ) )  e.  RR )
212101, 211sylan2 474 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( vol* `  A )  +  ( B  /  ( 2 ^ n ) ) )  e.  RR )
213 eqid 2457 . . . . . . . . . . . 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 22010 . . . . . . . . . . 11  |-  ( ( F `  n ) : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  S : NN --> ( 0 [,) +oo ) )
216190, 215syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  S : NN --> ( 0 [,) +oo ) )
217 frn 5743 . . . . . . . . . 10  |-  ( S : NN --> ( 0 [,) +oo )  ->  ran  S  C_  ( 0 [,) +oo ) )
218216, 217syl 16 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S 
C_  ( 0 [,) +oo ) )
219 icossxr 11634 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  RR*
220218, 219syl6ss 3511 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S 
C_  RR* )
221 supxrcl 11531 . . . . . . . 8  |-  ( ran 
S  C_  RR*  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
222220, 221syl 16 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR* )
223 mnfxr 11348 . . . . . . . . 9  |- -oo  e.  RR*
224223a1i 11 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  -> -oo  e.  RR* )
225103rexrd 9660 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol* `  A )  e.  RR* )
226 mnflt 11358 . . . . . . . . 9  |-  ( ( vol* `  A
)  e.  RR  -> -oo 
<  ( vol* `  A ) )
227103, 226syl 16 . . . . . . . 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 474 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  A  C_ 
U. ran  ( (,)  o.  ( F `  n
) ) )
230214ovollb 22016 . . . . . . . . 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 661 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol* `  A )  <_  sup ( ran  S ,  RR* ,  <  )
)
232224, 225, 222, 227, 231xrltletrd 11389 . . . . . . 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 474 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  <_  (
( vol* `  A )  +  ( B  /  ( 2 ^ n ) ) ) )
235 xrre 11395 . . . . . . 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 1229 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR )
237 1zzd 10916 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  1  e.  ZZ )
238213ovolfsval 22008 . . . . . . . . 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 471 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( ( abs  o.  -  )  o.  ( F `  n )
) `  i )  =  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) ) )
240213ovolfsf 22009 . . . . . . . . . . . . 13  |-  ( ( F `  n ) : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  (
( abs  o.  -  )  o.  ( F `  n
) ) : NN --> ( 0 [,) +oo ) )
241190, 240syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( abs  o.  -  )  o.  ( F `  n
) ) : NN --> ( 0 [,) +oo ) )
242241ffvelrnda 6032 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( ( abs  o.  -  )  o.  ( F `  n )
) `  i )  e.  ( 0 [,) +oo ) )
243239, 242eqeltrrd 2546 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  e.  ( 0 [,) +oo ) )
244 elrege0 11652 . . . . . . . . . 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 459 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  e.  RR )
247245simprd 463 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  0  <_  ( ( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) ) )
248 supxrub 11541 . . . . . . . . . . . . . . 15  |-  ( ( ran  S  C_  RR*  /\  z  e.  ran  S )  -> 
z  <_  sup ( ran  S ,  RR* ,  <  ) )
249220, 248sylan 471 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  z  e.  ran  S )  -> 
z  <_  sup ( ran  S ,  RR* ,  <  ) )
250249ralrimiva 2871 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  A. z  e.  ran  S  z  <_  sup ( ran  S ,  RR* ,  <  ) )
251 breq2 4460 . . . . . . . . . . . . . . 15  |-  ( x  =  sup ( ran 
S ,  RR* ,  <  )  ->  ( z  <_  x 
<->  z  <_  sup ( ran  S ,  RR* ,  <  ) ) )
252251ralbidv 2896 . . . . . . . . . . . . . 14  |-  ( x  =  sup ( ran 
S ,  RR* ,  <  )  ->  ( A. z  e.  ran  S  z  <_  x 
<-> 
A. z  e.  ran  S  z  <_  sup ( ran  S ,  RR* ,  <  ) ) )
253252rspcev 3210 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  E. x  e.  RR  A. z  e. 
ran  S  z  <_  x )
255 ffn 5737 . . . . . . . . . . . . . . 15  |-  ( S : NN --> ( 0 [,) +oo )  ->  S  Fn  NN )
256216, 255syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  S  Fn  NN )
257 breq1 4459 . . . . . . . . . . . . . . 15  |-  ( z  =  ( S `  k )  ->  (
z  <_  x  <->  ( S `  k )  <_  x
) )
258257ralrn 6035 . . . . . . . . . . . . . 14  |-  ( S  Fn  NN  ->  ( A. z  e.  ran  S  z  <_  x  <->  A. k  e.  NN  ( S `  k )  <_  x
) )
259256, 258syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( A. z  e.  ran  S  z  <_  x  <->  A. k  e.  NN  ( S `  k )  <_  x
) )
260259rexbidv 2968 . . . . . . . . . . . 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 13670 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  S  ~~>  sup ( ran  S ,  RR ,  <  ) )
263214, 262syl5eqbrr 4490 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  ( F `  n )
) )  ~~>  sup ( ran  S ,  RR ,  <  ) )
264 climrel 13327 . . . . . . . . . 10  |-  Rel  ~~>
265264releldmi 5249 . . . . . . . . 9  |-  (  seq 1 (  +  , 
( ( abs  o.  -  )  o.  ( F `  n )
) )  ~~>  sup ( ran  S ,  RR ,  <  )  ->  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  ( F `  n ) ) )  e.  dom  ~~>  )
266263, 265syl 16 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  ( F `  n )
) )  e.  dom  ~~>  )
26777, 237, 186, 198, 239, 246, 247, 266isumless 13669 . . . . . . 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 13671 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  NN  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  =  sup ( ran 
S ,  RR ,  <  ) )
269 rge0ssre 11653 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
270218, 269syl6ss 3511 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S 
C_  RR )
271 1nn 10567 . . . . . . . . . . . 12  |-  1  e.  NN
272 fdm 5741 . . . . . . . . . . . . 13  |-  ( S : NN --> ( 0 [,) +oo )  ->  dom  S  =  NN )
273216, 272syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  dom  S  =  NN )
274271, 273syl5eleqr 2552 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  1  e.  dom  S )
275 ne0i 3799 . . . . . . . . . . 11  |-  ( 1  e.  dom  S  ->  dom  S  =/=  (/) )
276274, 275syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  dom  S  =/=  (/) )
277 dm0rn0 5229 . . . . . . . . . . 11  |-  ( dom 
S  =  (/)  <->  ran  S  =  (/) )
278277necon3bii 2725 . . . . . . . . . 10  |-  ( dom 
S  =/=  (/)  <->  ran  S  =/=  (/) )
279276, 278sylib 196 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S  =/=  (/) )
280 supxrre 11544 . . . . . . . . 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 1228 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  S ,  RR ,  <  ) )
282268, 281eqtr4d 2501 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  NN  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  =  sup ( ran 
S ,  RR* ,  <  ) )
283267, 282breqtrd 4480 . . . . . 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 9756 . . . . 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 13625 . . . 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 3112 . . . . . . . . . . 11  |-  i  e. 
_V
287127, 286op1std 6809 . . . . . . . . . 10  |-  ( j  =  <. n ,  i
>.  ->  ( 1st `  j
)  =  n )
288287fveq2d 5876 . . . . . . . . 9  |-  ( j  =  <. n ,  i
>.  ->  ( F `  ( 1st `  j ) )  =  ( F `
 n ) )
289127, 286op2ndd 6810 . . . . . . . . 9  |-  ( j  =  <. n ,  i
>.  ->  ( 2nd `  j
)  =  i )
290288, 289fveq12d 5878 . . . . . . . 8  |-  ( j  =  <. n ,  i
>.  ->  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) )  =  ( ( F `
 n ) `  i ) )
291290fveq2d 5876 . . . . . . 7  |-  ( j  =  <. n ,  i
>.  ->  ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  =  ( 2nd `  (
( F `  n
) `  i )
) )
292290fveq2d 5876 . . . . . . 7  |-  ( j  =  <. n ,  i
>.  ->  ( 1st `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  =  ( 1st `  (
( F `  n
) `  i )
) )
293291, 292oveq12d 6314 . . . . . 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 9639 . . . . . 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 13598 . . . . 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 13535 . . . . 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 2498 . . . 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 9639 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol* `  A )  e.  CC )
299113recnd 9639 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( B  /  ( 2 ^ n ) )  e.  CC )
300100, 298, 299fsumadd 13573 . . . 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 4485 . . 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 10916 . . . . . . . . 9  |-  ( ph  ->  1  e.  ZZ )
303 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  NN )
304 ovoliun.g . . . . . . . . . . . 12  |-  G  =  ( n  e.  NN  |->  ( vol* `  A
) )
305304fvmpt2 5964 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  ( vol* `  A
)  e.  RR )  ->  ( G `  n )  =  ( vol* `  A
) )
306303, 102, 305syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  =  ( vol* `  A ) )
307306, 102eqeltrd 2545 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
30877, 302, 307serfre 12139 . . . . . . . 8  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> RR )
309 ovoliun.t . . . . . . . . 9  |-  T  =  seq 1 (  +  ,  G )
310309feq1i 5729 . . . . . . . 8  |-  ( T : NN --> RR  <->  seq 1
(  +  ,  G
) : NN --> RR )
311308, 310sylibr 212 . . . . . . 7  |-  ( ph  ->  T : NN --> RR )
312 frn 5743 . . . . . . 7  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
313311, 312syl 16 . . . . . 6  |-  ( ph  ->  ran  T  C_  RR )
314 ressxr 9654 . . . . . 6  |-  RR  C_  RR*
315313, 314syl6ss 3511 . . . . 5  |-  ( ph  ->  ran  T  C_  RR* )
316101, 306sylan2 474 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( G `  n )  =  ( vol* `  A ) )
317 1red 9628 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
318 ffvelrn 6030 . . . . . . . . . . . . 13  |-  ( ( J : NN --> ( NN 
X.  NN )  /\  1  e.  NN )  ->  ( J `  1
)  e.  ( NN 
X.  NN ) )
31923, 271, 318sylancl 662 . . . . . . . . . . . 12  |-  ( ph  ->  ( J `  1
)  e.  ( NN 
X.  NN ) )
320 xp1st 6829 . . . . . . . . . . . 12  |-  ( ( J `  1 )  e.  ( NN  X.  NN )  ->  ( 1st `  ( J `  1
) )  e.  NN )
321319, 320syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  e.  NN )
322321nnred 10571 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  e.  RR )
323153zred 10990 . . . . . . . . . 10  |-  ( ph  ->  L  e.  RR )
324321nnge1d 10599 . . . . . . . . . 10  |-  ( ph  ->  1  <_  ( 1st `  ( J `  1
) ) )
325 eluzfz1 11718 . . . . . . . . . . . 12  |-  ( K  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... K
) )
32678, 325syl 16 . . . . . . . . . . 11  |-  ( ph  ->  1  e.  ( 1 ... K ) )
327 fveq2 5872 . . . . . . . . . . . . . 14  |-  ( w  =  1  ->  ( J `  w )  =  ( J ` 
1 ) )
328327fveq2d 5876 . . . . . . . . . . . . 13  |-  ( w  =  1  ->  ( 1st `  ( J `  w ) )  =  ( 1st `  ( J `  1 )
) )
329328breq1d 4466 . . . . . . . . . . . 12  |-  ( w  =  1  ->  (
( 1st `  ( J `  w )
)  <_  L  <->  ( 1st `  ( J `  1
) )  <_  L
) )
330329rspcv 3206 . . . . . . . . . . 11  |-  ( 1  e.  ( 1 ... K )  ->  ( A. w  e.  (
1 ... K ) ( 1st `  ( J `
 w ) )  <_  L  ->  ( 1st `  ( J ` 
1 ) )  <_  L ) )
331326, 143, 330sylc 60 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  <_  L )
332317, 322, 323, 324, 331letrd 9756 . . . . . . . . 9  |-  ( ph  ->  1  <_  L )
333 elnnz1 10911 . . . . . . . . 9  |-  ( L  e.  NN  <->  ( L  e.  ZZ  /\  1  <_  L ) )
334153, 332, 333sylanbrc 664 . . . . . . . 8  |-  ( ph  ->  L  e.  NN )
335334, 77syl6eleq 2555 . . . . . . 7  |-  ( ph  ->  L  e.  ( ZZ>= ` 
1 ) )
336316, 335, 298fsumser 13564 . . . . . 6  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  =  (  seq 1 (  +  ,  G ) `  L
) )
337 seqfn 12122 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  seq 1 (  +  ,  G )  Fn  ( ZZ>=
`  1 ) )
338302, 337syl 16 . . . . . . . 8  |-  ( ph  ->  seq 1 (  +  ,  G )  Fn  ( ZZ>= `  1 )
)
339 fnfvelrn 6029 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  G )  Fn  ( ZZ>= `  1 )  /\  L  e.  ( ZZ>=
`  1 ) )  ->  (  seq 1
(  +  ,  G
) `  L )  e.  ran  seq 1 (  +  ,  G ) )
340338, 335, 339syl2anc 661 . . . . . . 7  |-  ( ph  ->  (  seq 1 (  +  ,  G ) `
 L )  e. 
ran  seq 1 (  +  ,  G ) )
341309rneqi 5239 . . . . . . 7  |-  ran  T  =  ran  seq 1 (  +  ,  G )
342340, 341syl6eleqr 2556 . . . . . 6  |-  ( ph  ->  (  seq 1 (  +  ,  G ) `
 L )  e. 
ran  T )
343336, 342eqeltrd 2545 . . . . 5  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  e.  ran  T
)
344 supxrub 11541 . . . . 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 661 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
346106recnd 9639 . . . . . 6  |-  ( ph  ->  B  e.  CC )
347 geo2sum 13694 . . . . . 6  |-  ( ( L  e.  NN  /\  B  e.  CC )  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  =  ( B  -  ( B  /  (
2 ^ L ) ) ) )
348334, 346, 347syl2anc 661 . . . . 5  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  =  ( B  -  ( B  /  (
2 ^ L ) ) ) )
349334nnnn0d 10873 . . . . . . . . . 10  |-  ( ph  ->  L  e.  NN0 )
350 nnexpcl 12182 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  L  e.  NN0 )  -> 
( 2 ^ L
)  e.  NN )
351107, 349, 350sylancr 663 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ L
)  e.  NN )
352351nnrpd 11280 . . . . . . . 8  |-  ( ph  ->  ( 2 ^ L
)  e.  RR+ )
353105, 352rpdivcld 11298 . . . . . . 7  |-  ( ph  ->  ( B  /  (
2 ^ L ) )  e.  RR+ )
354353rpge0d 11285 . . . . . 6  |-  ( ph  ->  0  <_  ( B  /  ( 2 ^ L ) ) )
355106, 351nndivred 10605 . . . . . . 7  |-  ( ph  ->  ( B  /  (
2 ^ L ) )  e.  RR )
356106, 355subge02d 10165 . . . . . 6  |-  ( ph  ->  ( 0  <_  ( B  /  ( 2 ^ L ) )  <->  ( B  -  ( B  / 
( 2 ^ L
) ) )  <_  B ) )
357354, 356mpbid 210 . . . . 5  |-  ( ph  ->  ( B  -  ( B  /  ( 2 ^ L ) ) )  <_  B )
358348, 357eqbrtrd 4476 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  <_  B )
359104, 114, 116, 106, 345, 358le2addd 10191 . . 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 9756 . 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 4478 1  |-  ( ph  ->  ( U `  K
)  <_  ( sup ( ran  T ,  RR* ,  <  )  +  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   <.cop 4038   U.cuni 4251   U_ciun 4332   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011    o. ccom 5012   Rel wrel 5013    Fn wfn 5589   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798    ^m cmap 7438    ~~ cen 7532   Fincfn 7535   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   RR+crp 11245   (,)cioo 11554   [,)cico 11556   ...cfz 11697    seqcseq 12110   ^cexp 12169   abscabs 13079    ~~> cli 13319   sum_csu 13520   vol*covol 22000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ioo 11558  df-ico 11560  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-ovol 22002
This theorem is referenced by:  ovoliunlem2  22040
  Copyright terms: Public domain W3C validator