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Theorem ovoliunlem1 21116
Description: Lemma for ovoliun 21119. (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 5798 . . . . . . . . 9  |-  ( j  =  ( J `  m )  ->  ( 1st `  j )  =  ( 1st `  ( J `  m )
) )
21fveq2d 5802 . . . . . . . 8  |-  ( j  =  ( J `  m )  ->  ( F `  ( 1st `  j ) )  =  ( F `  ( 1st `  ( J `  m ) ) ) )
3 fveq2 5798 . . . . . . . 8  |-  ( j  =  ( J `  m )  ->  ( 2nd `  j )  =  ( 2nd `  ( J `  m )
) )
42, 3fveq12d 5804 . . . . . . 7  |-  ( j  =  ( J `  m )  ->  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) )  =  ( ( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )
54fveq2d 5802 . . . . . 6  |-  ( j  =  ( J `  m )  ->  ( 2nd `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  =  ( 2nd `  ( ( F `  ( 1st `  ( J `
 m ) ) ) `  ( 2nd `  ( J `  m
) ) ) ) )
64fveq2d 5802 . . . . . 6  |-  ( j  =  ( J `  m )  ->  ( 1st `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  =  ( 1st `  ( ( F `  ( 1st `  ( J `
 m ) ) ) `  ( 2nd `  ( J `  m
) ) ) ) )
75, 6oveq12d 6217 . . . . 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 11911 . . . . 5  |-  ( ph  ->  ( 1 ... K
)  e.  Fin )
9 ovoliun.j . . . . . . 7  |-  ( ph  ->  J : NN -1-1-onto-> ( NN  X.  NN ) )
10 f1of1 5747 . . . . . . 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 11594 . . . . . . 7  |-  ( m  e.  ( 1 ... K )  ->  m  e.  NN )
1312ssriv 3467 . . . . . 6  |-  ( 1 ... K )  C_  NN
14 f1ores 5762 . . . . . 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 5812 . . . . . 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 3678 . . . . . . . . 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 5287 . . . . . . . . . . . . . . 15  |-  ( J
" ( 1 ... K ) )  C_  ran  J
22 f1of 5748 . . . . . . . . . . . . . . . . 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 5672 . . . . . . . . . . . . . . . 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 3474 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  ( NN  X.  NN ) )
2726sselda 3463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  j  e.  ( NN  X.  NN ) )
28 xp1st 6715 . . . . . . . . . . . . 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 5952 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( F `  ( 1st `  j
) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
31 reex 9483 . . . . . . . . . . . . . 14  |-  RR  e.  _V
3231, 31xpex 6617 . . . . . . . . . . . . 13  |-  ( RR 
X.  RR )  e. 
_V
3332inex2 4541 . . . . . . . . . . . 12  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
34 nnex 10438 . . . . . . . . . . . 12  |-  NN  e.  _V
3533, 34elmap 7350 . . . . . . . . . . 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 6716 . . . . . . . . . . 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 5952 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
4018, 39sseldi 3461 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) )  e.  ( RR 
X.  RR ) )
41 xp2nd 6716 . . . . . . . 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 6715 . . . . . . . 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 9886 . . . . . 6  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( ( 2nd `  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  e.  RR )
4645recnd 9522 . . . . 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 13317 . . . 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 5951 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  ( J `
 k )  e.  ( NN  X.  NN ) )
50 xp1st 6715 . . . . . . . . . . . 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 5952 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 ( 1st `  ( J `  k )
) )  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) )
5333, 34elmap 7350 . . . . . . . . . 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 6716 . . . . . . . . . 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 5952 . . . . . . . 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 5975 . . . . . . 7  |-  ( ph  ->  H : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
60 eqid 2454 . . . . . . . 8  |-  ( ( abs  o.  -  )  o.  H )  =  ( ( abs  o.  -  )  o.  H )
6160ovolfsval 21085 . . . . . . 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 5798 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( J `  k )  =  ( J `  m ) )
6564fveq2d 5802 . . . . . . . . . . . 12  |-  ( k  =  m  ->  ( 1st `  ( J `  k ) )  =  ( 1st `  ( J `  m )
) )
6665fveq2d 5802 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( F `  ( 1st `  ( J `  k
) ) )  =  ( F `  ( 1st `  ( J `  m ) ) ) )
6764fveq2d 5802 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( 2nd `  ( J `  k ) )  =  ( 2nd `  ( J `  m )
) )
6866, 67fveq12d 5804 . . . . . . . . . 10  |-  ( k  =  m  ->  (
( F `  ( 1st `  ( J `  k ) ) ) `
 ( 2nd `  ( J `  k )
) )  =  ( ( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) )
69 fvex 5808 . . . . . . . . . 10  |-  ( ( F `  ( 1st `  ( J `  m
) ) ) `  ( 2nd `  ( J `
 m ) ) )  e.  _V
7068, 58, 69fvmpt 5882 . . . . . . . . 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 5802 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 2nd `  ( H `  m ) )  =  ( 2nd `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) ) )
7371fveq2d 5802 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( 1st `  ( H `  m ) )  =  ( 1st `  (
( F `  ( 1st `  ( J `  m ) ) ) `
 ( 2nd `  ( J `  m )
) ) ) )
7472, 73oveq12d 6217 . . . . . 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 2495 . . . . 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 11006 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
7876, 77syl6eleq 2552 . . . . 5  |-  ( ph  ->  K  e.  ( ZZ>= ` 
1 ) )
79 ffvelrn 5949 . . . . . . . . . . 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 3461 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  ( H `  m )  e.  ( RR  X.  RR ) )
82 xp2nd 6716 . . . . . . . . 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 6715 . . . . . . . . 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 9886 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( 2nd `  ( H `  m )
)  -  ( 1st `  ( H `  m
) ) )  e.  RR )
8786recnd 9522 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... K
) )  ->  (
( 2nd `  ( H `  m )
)  -  ( 1st `  ( H `  m
) ) )  e.  CC )
8874, 87eqeltrrd 2543 . . . . 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 13324 . . . 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 2495 . . 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 5799 . . 3  |-  ( U `
 K )  =  (  seq 1 (  +  ,  ( ( abs  o.  -  )  o.  H ) ) `  K )
9390, 92syl6eqr 2513 . 2  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  =  ( U `  K
) )
94 f1oeng 7437 . . . . . . 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 7469 . . . . 5  |-  ( ph  ->  ( J " (
1 ... K ) ) 
~~  ( 1 ... K ) )
97 enfii 7640 . . . . 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 13328 . . 3  |-  ( ph  -> 
sum_ j  e.  ( J " ( 1 ... K ) ) ( ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  -  ( 1st `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) ) )  e.  RR )
100 fzfid 11911 . . . . 5  |-  ( ph  ->  ( 1 ... L
)  e.  Fin )
101 elfznn 11594 . . . . . 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 13328 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  e.  RR )
105 ovoliun.b . . . . . . 7  |-  ( ph  ->  B  e.  RR+ )
106105rpred 11137 . . . . . 6  |-  ( ph  ->  B  e.  RR )
107 2nn 10589 . . . . . . . 8  |-  2  e.  NN
108 nnnn0 10696 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  NN0 )
109 nnexpcl 11994 . . . . . . . 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 10467 . . . . . 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 13328 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  e.  RR )
115104, 114readdcld 9523 . . 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 9523 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  +  B )  e.  RR )
118 relxp 5054 . . . . . . . . . . . . . . 15  |-  Rel  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )
119 relres 5245 . . . . . . . . . . . . . . 15  |-  Rel  (
( J " (
1 ... K ) )  |`  { n } )
120 opelxp 4976 . . . . . . . . . . . . . . . 16  |-  ( <.
x ,  y >.  e.  ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  <-> 
( x  e.  {
n }  /\  y  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )
121 vex 3079 . . . . . . . . . . . . . . . . . 18  |-  y  e. 
_V
122121opelres 5223 . . . . . . . . . . . . . . . . 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 4009 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  { n }  ->  x  =  n )
125124opeq1d 4172 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  { n }  -> 
<. x ,  y >.  =  <. n ,  y
>. )
126125eleq1d 2523 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  { n }  ->  ( <. x ,  y
>.  e.  ( J "
( 1 ... K
) )  <->  <. n ,  y >.  e.  ( J " ( 1 ... K ) ) ) )
127 vex 3079 . . . . . . . . . . . . . . . . . . . 20  |-  n  e. 
_V
128127, 121elimasn 5301 . . . . . . . . . . . . . . . . . . 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 5041 . . . . . . . . . . . . . 14  |-  ( { n }  X.  (
( J " (
1 ... K ) )
" { n }
) )  =  ( ( J " (
1 ... K ) )  |`  { n } )
134 df-res 4959 . . . . . . . . . . . . . 14  |-  ( ( J " ( 1 ... K ) )  |`  { n } )  =  ( ( J
" ( 1 ... K ) )  i^i  ( { n }  X.  _V ) )
135133, 134eqtri 2483 . . . . . . . . . . . . 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 4299 . . . . . . . . . . 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 4341 . . . . . . . . . . 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 2511 . . . . . . . . . 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 5054 . . . . . . . . . . . . . 14  |-  Rel  ( NN  X.  NN )
141 relss 5034 . . . . . . . . . . . . . 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 5666 . . . . . . . . . . . . . . . . . . . . 21  |-  ( J : NN --> ( NN 
X.  NN )  ->  J  Fn  NN )
14523, 144syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  J  Fn  NN )
146 fveq2 5798 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( j  =  ( J `  w )  ->  ( 1st `  j )  =  ( 1st `  ( J `  w )
) )
147146breq1d 4409 . . . . . . . . . . . . . . . . . . . . 21  |-  ( j  =  ( J `  w )  ->  (
( 1st `  j
)  <_  L  <->  ( 1st `  ( J `  w
) )  <_  L
) )
148147ralima 6065 . . . . . . . . . . . . . . . . . . . 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 2918 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  j  e.  ( J " ( 1 ... K ) ) )  ->  ( 1st `  j )  <_  L
)
15229, 77syl6eleq 2552 . . . . . . . . . . . . . . . . . 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 11561 . . . . . . . . . . . . . . . . . 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 2829 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. j  e.  ( J " ( 1 ... K ) ) ( 1st `  j
)  e.  ( 1 ... L ) )
159 vex 3079 . . . . . . . . . . . . . . . . . 18  |-  x  e. 
_V
160159, 121op1std 6696 . . . . . . . . . . . . . . . . 17  |-  ( j  =  <. x ,  y
>.  ->  ( 1st `  j
)  =  x )
161160eleq1d 2523 . . . . . . . . . . . . . . . 16  |-  ( j  =  <. x ,  y
>.  ->  ( ( 1st `  j )  e.  ( 1 ... L )  <-> 
x  e.  ( 1 ... L ) ) )
162161rspccv 3174 . . . . . . . . . . . . . . 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 4976 . . . . . . . . . . . . . . 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 4332 . . . . . . . . . . . . . . . 16  |-  U_ n  e.  ( 1 ... L
) { n }  =  ( 1 ... L )
167166eleq2i 2532 . . . . . . . . . . . . . . 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 5039 . . . . . . . . . . . 12  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  ( U_ n  e.  ( 1 ... L
) { n }  X.  _V ) )
171 xpiundir 5001 . . . . . . . . . . . 12  |-  ( U_ n  e.  ( 1 ... L ) { n }  X.  _V )  =  U_ n  e.  ( 1 ... L
) ( { n }  X.  _V )
172170, 171syl6sseq 3509 . . . . . . . . . . 11  |-  ( ph  ->  ( J " (
1 ... K ) ) 
C_  U_ n  e.  ( 1 ... L ) ( { n }  X.  _V ) )
173 df-ss 3449 . . . . . . . . . . 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 2495 . . . . . . . . 9  |-  ( ph  ->  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  =  ( J "
( 1 ... K
) ) )
176175, 98eqeltrd 2542 . . . . . . . 8  |-  ( ph  ->  U_ n  e.  ( 1 ... L ) ( { n }  X.  ( ( J "
( 1 ... K
) ) " {
n } ) )  e.  Fin )
177 ssiun2 4320 . . . . . . . 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 7643 . . . . . . . 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 6771 . . . . . . . . . 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 7437 . . . . . . . . 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 7469 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( J " (
1 ... K ) )
" { n }
)  ~~  ( {
n }  X.  (
( J " (
1 ... K ) )
" { n }
) ) )
185 enfii 7640 . . . . . . 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 5949 . . . . . . . . . . . . . 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 7350 . . . . . . . . . . . . 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 5287 . . . . . . . . . . . . . 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 5175 . . . . . . . . . . . . . . . 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 5378 . . . . . . . . . . . . . . 15  |-  ran  ( NN  X.  NN )  =  NN
197195, 196syl6sseq 3509 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  ( J " ( 1 ... K ) ) 
C_  NN )
198192, 197syl5ss 3474 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  (
( J " (
1 ... K ) )
" { n }
)  C_  NN )
199198sseld 3462 . . . . . . . . . . . 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 5952 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( F `  n ) `  i )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
20218, 201sseldi 3461 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( F `  n ) `  i )  e.  ( RR  X.  RR ) )
203 xp2nd 6716 . . . . . . . . 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 6715 . . . . . . . . 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 9886 . . . . . . 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 13328 . . . . 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 9523 . . . . . 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 2454 . . . . . . . . . . . 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 21087 . . . . . . . . . . 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 5672 . . . . . . . . . 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 11490 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  RR*
220218, 219syl6ss 3475 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S 
C_  RR* )
221 supxrcl 11387 . . . . . . . 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 11204 . . . . . . . . 9  |- -oo  e.  RR*
224223a1i 11 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  -> -oo  e.  RR* )
225103rexrd 9543 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol* `  A )  e.  RR* )
226 mnflt 11214 . . . . . . . . 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 21093 . . . . . . . . 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 11245 . . . . . . 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 11251 . . . . . . 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 1220 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  e.  RR )
237 1zzd 10787 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  1  e.  ZZ )
238213ovolfsval 21085 . . . . . . . . 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 21086 . . . . . . . . . . . . 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 5951 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( ( abs  o.  -  )  o.  ( F `  n )
) `  i )  e.  ( 0 [,) +oo ) )
243239, 242eqeltrrd 2543 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( 1 ... L
) )  /\  i  e.  NN )  ->  (
( 2nd `  (
( F `  n
) `  i )
)  -  ( 1st `  ( ( F `  n ) `  i
) ) )  e.  ( 0 [,) +oo ) )
244 elrege0 11508 . . . . . . . . . 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 11397 . . . . . . . . . . . . . . 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 2829 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  A. z  e.  ran  S  z  <_  sup ( ran  S ,  RR* ,  <  ) )
251 breq2 4403 . . . . . . . . . . . . . . 15  |-  ( x  =  sup ( ran 
S ,  RR* ,  <  )  ->  ( z  <_  x 
<->  z  <_  sup ( ran  S ,  RR* ,  <  ) ) )
252251ralbidv 2845 . . . . . . . . . . . . . 14  |-  ( x  =  sup ( ran 
S ,  RR* ,  <  )  ->  ( A. z  e.  ran  S  z  <_  x 
<-> 
A. z  e.  ran  S  z  <_  sup ( ran  S ,  RR* ,  <  ) ) )
253252rspcev 3177 . . . . . . . . . . . . 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 5666 . . . . . . . . . . . . . . 15  |-  ( S : NN --> ( 0 [,) +oo )  ->  S  Fn  NN )
256216, 255syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  S  Fn  NN )
257 breq1 4402 . . . . . . . . . . . . . . 15  |-  ( z  =  ( S `  k )  ->  (
z  <_  x  <->  ( S `  k )  <_  x
) )
258257ralrn 5954 . . . . . . . . . . . . . 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 2864 . . . . . . . . . . . 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 13426 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  S  ~~>  sup ( ran  S ,  RR ,  <  ) )
263214, 262syl5eqbrr 4433 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  seq 1 (  +  , 
( ( abs  o.  -  )  o.  ( F `  n )
) )  ~~>  sup ( ran  S ,  RR ,  <  ) )
264 climrel 13087 . . . . . . . . . 10  |-  Rel  ~~>
265264releldmi 5183 . . . . . . . . 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 13425 . . . . . . 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 13427 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  NN  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  =  sup ( ran 
S ,  RR ,  <  ) )
269 0re 9496 . . . . . . . . . . 11  |-  0  e.  RR
270 pnfxr 11202 . . . . . . . . . . 11  |- +oo  e.  RR*
271 icossre 11486 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
272269, 270, 271mp2an 672 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
273218, 272syl6ss 3475 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S 
C_  RR )
274 1nn 10443 . . . . . . . . . . . 12  |-  1  e.  NN
275 fdm 5670 . . . . . . . . . . . . 13  |-  ( S : NN --> ( 0 [,) +oo )  ->  dom  S  =  NN )
276216, 275syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  dom  S  =  NN )
277274, 276syl5eleqr 2549 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  1  e.  dom  S )
278 ne0i 3750 . . . . . . . . . . 11  |-  ( 1  e.  dom  S  ->  dom  S  =/=  (/) )
279277, 278syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  dom  S  =/=  (/) )
280 dm0rn0 5163 . . . . . . . . . . 11  |-  ( dom 
S  =  (/)  <->  ran  S  =  (/) )
281280necon3bii 2719 . . . . . . . . . 10  |-  ( dom 
S  =/=  (/)  <->  ran  S  =/=  (/) )
282279, 281sylib 196 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ran  S  =/=  (/) )
283 supxrre 11400 . . . . . . . . 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 ,  <  ) )
284273, 282, 254, 283syl3anc 1219 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sup ( ran  S ,  RR* ,  <  )  =  sup ( ran  S ,  RR ,  <  ) )
285268, 284eqtr4d 2498 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  sum_ i  e.  NN  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) )  =  sup ( ran 
S ,  RR* ,  <  ) )
286267, 285breqtrd 4423 . . . . . 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* ,  <  ) )
287209, 236, 212, 286, 234letrd 9638 . . . . 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 ) ) ) )
288100, 209, 212, 287fsumle 13379 . . . 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 ) ) ) )
289 vex 3079 . . . . . . . . . . 11  |-  i  e. 
_V
290127, 289op1std 6696 . . . . . . . . . 10  |-  ( j  =  <. n ,  i
>.  ->  ( 1st `  j
)  =  n )
291290fveq2d 5802 . . . . . . . . 9  |-  ( j  =  <. n ,  i
>.  ->  ( F `  ( 1st `  j ) )  =  ( F `
 n ) )
292127, 289op2ndd 6697 . . . . . . . . 9  |-  ( j  =  <. n ,  i
>.  ->  ( 2nd `  j
)  =  i )
293291, 292fveq12d 5804 . . . . . . . 8  |-  ( j  =  <. n ,  i
>.  ->  ( ( F `
 ( 1st `  j
) ) `  ( 2nd `  j ) )  =  ( ( F `
 n ) `  i ) )
294293fveq2d 5802 . . . . . . 7  |-  ( j  =  <. n ,  i
>.  ->  ( 2nd `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  =  ( 2nd `  (
( F `  n
) `  i )
) )
295293fveq2d 5802 . . . . . . 7  |-  ( j  =  <. n ,  i
>.  ->  ( 1st `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) )  =  ( 1st `  (
( F `  n
) `  i )
) )
296294, 295oveq12d 6217 . . . . . 6  |-  ( j  =  <. n ,  i
>.  ->  ( ( 2nd `  ( ( F `  ( 1st `  j ) ) `  ( 2nd `  j ) ) )  -  ( 1st `  (
( F `  ( 1st `  j ) ) `
 ( 2nd `  j
) ) ) )  =  ( ( 2nd `  ( ( F `  n ) `  i
) )  -  ( 1st `  ( ( F `
 n ) `  i ) ) ) )
297207recnd 9522 . . . . . 6  |-  ( (
ph  /\  ( n  e.  ( 1 ... L
)  /\  i  e.  ( ( J "
( 1 ... K
) ) " {
n } ) ) )  ->  ( ( 2nd `  ( ( F `
 n ) `  i ) )  -  ( 1st `  ( ( F `  n ) `
 i ) ) )  e.  CC )
298296, 100, 186, 297fsum2d 13355 . . . . 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 ) ) ) ) )
299175sumeq1d 13295 . . . . 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 ) ) ) ) )
300298, 299eqtrd 2495 . . . 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 ) ) ) ) )
301103recnd 9522 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( vol* `  A )  e.  CC )
302113recnd 9522 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( B  /  ( 2 ^ n ) )  e.  CC )
303100, 301, 302fsumadd 13332 . . . 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 ) ) ) )
304288, 300, 3033brtr3d 4428 . . 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 ) ) ) )
305 1zzd 10787 . . . . . . . . 9  |-  ( ph  ->  1  e.  ZZ )
306 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN )  ->  n  e.  NN )
307 ovoliun.g . . . . . . . . . . . 12  |-  G  =  ( n  e.  NN  |->  ( vol* `  A
) )
308307fvmpt2 5889 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  ( vol* `  A
)  e.  RR )  ->  ( G `  n )  =  ( vol* `  A
) )
309306, 102, 308syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  =  ( vol* `  A ) )
310309, 102eqeltrd 2542 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
31177, 305, 310serfre 11951 . . . . . . . 8  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> RR )
312 ovoliun.t . . . . . . . . 9  |-  T  =  seq 1 (  +  ,  G )
313312feq1i 5658 . . . . . . . 8  |-  ( T : NN --> RR  <->  seq 1
(  +  ,  G
) : NN --> RR )
314311, 313sylibr 212 . . . . . . 7  |-  ( ph  ->  T : NN --> RR )
315 frn 5672 . . . . . . 7  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
316314, 315syl 16 . . . . . 6  |-  ( ph  ->  ran  T  C_  RR )
317 ressxr 9537 . . . . . 6  |-  RR  C_  RR*
318316, 317syl6ss 3475 . . . . 5  |-  ( ph  ->  ran  T  C_  RR* )
319101, 309sylan2 474 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... L
) )  ->  ( G `  n )  =  ( vol* `  A ) )
320 1red 9511 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
321 ffvelrn 5949 . . . . . . . . . . . . 13  |-  ( ( J : NN --> ( NN 
X.  NN )  /\  1  e.  NN )  ->  ( J `  1
)  e.  ( NN 
X.  NN ) )
32223, 274, 321sylancl 662 . . . . . . . . . . . 12  |-  ( ph  ->  ( J `  1
)  e.  ( NN 
X.  NN ) )
323 xp1st 6715 . . . . . . . . . . . 12  |-  ( ( J `  1 )  e.  ( NN  X.  NN )  ->  ( 1st `  ( J `  1
) )  e.  NN )
324322, 323syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  e.  NN )
325324nnred 10447 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  e.  RR )
326153zred 10857 . . . . . . . . . 10  |-  ( ph  ->  L  e.  RR )
327324nnge1d 10474 . . . . . . . . . 10  |-  ( ph  ->  1  <_  ( 1st `  ( J `  1
) ) )
328 eluzfz1 11574 . . . . . . . . . . . 12  |-  ( K  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... K
) )
32978, 328syl 16 . . . . . . . . . . 11  |-  ( ph  ->  1  e.  ( 1 ... K ) )
330 fveq2 5798 . . . . . . . . . . . . . 14  |-  ( w  =  1  ->  ( J `  w )  =  ( J ` 
1 ) )
331330fveq2d 5802 . . . . . . . . . . . . 13  |-  ( w  =  1  ->  ( 1st `  ( J `  w ) )  =  ( 1st `  ( J `  1 )
) )
332331breq1d 4409 . . . . . . . . . . . 12  |-  ( w  =  1  ->  (
( 1st `  ( J `  w )
)  <_  L  <->  ( 1st `  ( J `  1
) )  <_  L
) )
333332rspcv 3173 . . . . . . . . . . 11  |-  ( 1  e.  ( 1 ... K )  ->  ( A. w  e.  (
1 ... K ) ( 1st `  ( J `
 w ) )  <_  L  ->  ( 1st `  ( J ` 
1 ) )  <_  L ) )
334329, 143, 333sylc 60 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  ( J `  1 )
)  <_  L )
335320, 325, 326, 327, 334letrd 9638 . . . . . . . . 9  |-  ( ph  ->  1  <_  L )
336 elnnz1 10782 . . . . . . . . 9  |-  ( L  e.  NN  <->  ( L  e.  ZZ  /\  1  <_  L ) )
337153, 335, 336sylanbrc 664 . . . . . . . 8  |-  ( ph  ->  L  e.  NN )
338337, 77syl6eleq 2552 . . . . . . 7  |-  ( ph  ->  L  e.  ( ZZ>= ` 
1 ) )
339319, 338, 301fsumser 13324 . . . . . 6  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  =  (  seq 1 (  +  ,  G ) `  L
) )
340 seqfn 11934 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  seq 1 (  +  ,  G )  Fn  ( ZZ>=
`  1 ) )
341305, 340syl 16 . . . . . . . 8  |-  ( ph  ->  seq 1 (  +  ,  G )  Fn  ( ZZ>= `  1 )
)
342 fnfvelrn 5948 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  G )  Fn  ( ZZ>= `  1 )  /\  L  e.  ( ZZ>=
`  1 ) )  ->  (  seq 1
(  +  ,  G
) `  L )  e.  ran  seq 1 (  +  ,  G ) )
343341, 338, 342syl2anc 661 . . . . . . 7  |-  ( ph  ->  (  seq 1 (  +  ,  G ) `
 L )  e. 
ran  seq 1 (  +  ,  G ) )
344312rneqi 5173 . . . . . . 7  |-  ran  T  =  ran  seq 1 (  +  ,  G )
345343, 344syl6eleqr 2553 . . . . . 6  |-  ( ph  ->  (  seq 1 (  +  ,  G ) `
 L )  e. 
ran  T )
346339, 345eqeltrd 2542 . . . . 5  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  e.  ran  T
)
347 supxrub 11397 . . . . 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* ,  <  ) )
348318, 346, 347syl2anc 661 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( vol* `  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
349106recnd 9522 . . . . . 6  |-  ( ph  ->  B  e.  CC )
350 geo2sum 13450 . . . . . 6  |-  ( ( L  e.  NN  /\  B  e.  CC )  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  =  ( B  -  ( B  /  (
2 ^ L ) ) ) )
351337, 349, 350syl2anc 661 . . . . 5  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  =  ( B  -  ( B  /  (
2 ^ L ) ) ) )
352337nnnn0d 10746 . . . . . . . . . 10  |-  ( ph  ->  L  e.  NN0 )
353 nnexpcl 11994 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  L  e.  NN0 )  -> 
( 2 ^ L
)  e.  NN )
354107, 352, 353sylancr 663 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ L
)  e.  NN )
355354nnrpd 11136 . . . . . . . 8  |-  ( ph  ->  ( 2 ^ L
)  e.  RR+ )
356105, 355rpdivcld 11154 . . . . . . 7  |-  ( ph  ->  ( B  /  (
2 ^ L ) )  e.  RR+ )
357356rpge0d 11141 . . . . . 6  |-  ( ph  ->  0  <_  ( B  /  ( 2 ^ L ) ) )
358106, 354nndivred 10480 . . . . . . 7  |-  ( ph  ->  ( B  /  (
2 ^ L ) )  e.  RR )
359106, 358subge02d 10041 . . . . . 6  |-  ( ph  ->  ( 0  <_  ( B  /  ( 2 ^ L ) )  <->  ( B  -  ( B  / 
( 2 ^ L
) ) )  <_  B ) )
360357, 359mpbid 210 . . . . 5  |-  ( ph  ->  ( B  -  ( B  /  ( 2 ^ L ) ) )  <_  B )
361351, 360eqbrtrd 4419 . . . 4  |-  ( ph  -> 
sum_ n  e.  (
1 ... L ) ( B  /  ( 2 ^ n ) )  <_  B )
362104, 114, 116, 106, 348, 361le2addd 10067 . . 3  |-  ( ph  ->  ( sum_ n  e.  ( 1 ... L ) ( vol* `  A )  +  sum_ n  e.  ( 1 ... L ) ( B  /  ( 2 ^ n ) ) )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  B ) )
36399, 115, 117, 304, 362letrd 9638 . 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 ) )
36493, 363eqbrtrrd 4421 1  |-  ( ph  ->  ( U `  K
)  <_  ( sup ( ran  T ,  RR* ,  <  )  +  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647   A.wral 2798   E.wrex 2799   _Vcvv 3076    i^i cin 3434    C_ wss 3435   (/)c0 3744   {csn 3984   <.cop 3990   U.cuni 4198   U_ciun 4278   class class class wbr 4399    |-> cmpt 4457    X. cxp 4945   dom cdm 4947   ran crn 4948    |` cres 4949   "cima 4950    o. ccom 4951   Rel wrel 4952    Fn wfn 5520   -->wf 5521   -1-1->wf1 5522   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6199   1stc1st 6684   2ndc2nd 6685    ^m cmap 7323    ~~ cen 7416   Fincfn 7419   supcsup 7800   CCcc 9390   RRcr 9391   0cc0 9392   1c1 9393    + caddc 9395   +oocpnf 9525   -oocmnf 9526   RR*cxr 9527    < clt 9528    <_ cle 9529    - cmin 9705    / cdiv 10103   NNcn 10432   2c2 10481   NN0cn0 10689   ZZcz 10756   ZZ>=cuz 10971   RR+crp 11101   (,)cioo 11410   [,)cico 11412   ...cfz 11553    seqcseq 11922   ^cexp 11981   abscabs 12840    ~~> cli 13079   sum_csu 13280   vol*covol 21077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-oi 7834  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-rp 11102  df-ioo 11414  df-ico 11416  df-fz 11554  df-fzo 11665  df-fl 11758  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-rlim 13084  df-sum 13281  df-ovol 21079
This theorem is referenced by:  ovoliunlem2  21117
  Copyright terms: Public domain W3C validator