| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ovollb2.1 |
|- S = seq 1 ( + , ( ( abs o. - ) o. F ) ) |
| 2 |
|
ovollb2.2 |
|- G = ( n e. NN |-> <. ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) , ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) >. ) |
| 3 |
|
ovollb2.3 |
|- T = seq 1 ( + , ( ( abs o. - ) o. G ) ) |
| 4 |
|
ovollb2.4 |
|- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) ) |
| 5 |
|
ovollb2.5 |
|- ( ph -> A C_ U. ran ( [,] o. F ) ) |
| 6 |
|
ovollb2.6 |
|- ( ph -> B e. RR+ ) |
| 7 |
|
ovollb2.7 |
|- ( ph -> sup ( ran S , RR* , < ) e. RR ) |
| 8 |
|
ovolficcss |
|- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> U. ran ( [,] o. F ) C_ RR ) |
| 9 |
4 8
|
syl |
|- ( ph -> U. ran ( [,] o. F ) C_ RR ) |
| 10 |
5 9
|
sstrd |
|- ( ph -> A C_ RR ) |
| 11 |
|
ovolcl |
|- ( A C_ RR -> ( vol* ` A ) e. RR* ) |
| 12 |
10 11
|
syl |
|- ( ph -> ( vol* ` A ) e. RR* ) |
| 13 |
|
ovolfcl |
|- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) ) |
| 14 |
4 13
|
sylan |
|- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) e. RR /\ ( 2nd ` ( F ` n ) ) e. RR /\ ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) ) |
| 15 |
14
|
simp1d |
|- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) e. RR ) |
| 16 |
6
|
rphalfcld |
|- ( ph -> ( B / 2 ) e. RR+ ) |
| 17 |
16
|
adantr |
|- ( ( ph /\ n e. NN ) -> ( B / 2 ) e. RR+ ) |
| 18 |
|
2nn |
|- 2 e. NN |
| 19 |
|
nnnn0 |
|- ( n e. NN -> n e. NN0 ) |
| 20 |
19
|
adantl |
|- ( ( ph /\ n e. NN ) -> n e. NN0 ) |
| 21 |
|
nnexpcl |
|- ( ( 2 e. NN /\ n e. NN0 ) -> ( 2 ^ n ) e. NN ) |
| 22 |
18 20 21
|
sylancr |
|- ( ( ph /\ n e. NN ) -> ( 2 ^ n ) e. NN ) |
| 23 |
22
|
nnrpd |
|- ( ( ph /\ n e. NN ) -> ( 2 ^ n ) e. RR+ ) |
| 24 |
17 23
|
rpdivcld |
|- ( ( ph /\ n e. NN ) -> ( ( B / 2 ) / ( 2 ^ n ) ) e. RR+ ) |
| 25 |
24
|
rpred |
|- ( ( ph /\ n e. NN ) -> ( ( B / 2 ) / ( 2 ^ n ) ) e. RR ) |
| 26 |
15 25
|
resubcld |
|- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) e. RR ) |
| 27 |
14
|
simp2d |
|- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) e. RR ) |
| 28 |
27 25
|
readdcld |
|- ( ( ph /\ n e. NN ) -> ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) e. RR ) |
| 29 |
15 24
|
ltsubrpd |
|- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) < ( 1st ` ( F ` n ) ) ) |
| 30 |
14
|
simp3d |
|- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) <_ ( 2nd ` ( F ` n ) ) ) |
| 31 |
27 24
|
ltaddrpd |
|- ( ( ph /\ n e. NN ) -> ( 2nd ` ( F ` n ) ) < ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) ) |
| 32 |
15 27 28 30 31
|
lelttrd |
|- ( ( ph /\ n e. NN ) -> ( 1st ` ( F ` n ) ) < ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) ) |
| 33 |
26 15 28 29 32
|
lttrd |
|- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) < ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) ) |
| 34 |
26 28 33
|
ltled |
|- ( ( ph /\ n e. NN ) -> ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) <_ ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) ) |
| 35 |
|
df-br |
|- ( ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) <_ ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) <-> <. ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) , ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) >. e. <_ ) |
| 36 |
34 35
|
sylib |
|- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) , ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) >. e. <_ ) |
| 37 |
26 28
|
opelxpd |
|- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) , ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) >. e. ( RR X. RR ) ) |
| 38 |
36 37
|
elind |
|- ( ( ph /\ n e. NN ) -> <. ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) , ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) >. e. ( <_ i^i ( RR X. RR ) ) ) |
| 39 |
38 2
|
fmptd |
|- ( ph -> G : NN --> ( <_ i^i ( RR X. RR ) ) ) |
| 40 |
|
eqid |
|- ( ( abs o. - ) o. G ) = ( ( abs o. - ) o. G ) |
| 41 |
40 3
|
ovolsf |
|- ( G : NN --> ( <_ i^i ( RR X. RR ) ) -> T : NN --> ( 0 [,) +oo ) ) |
| 42 |
39 41
|
syl |
|- ( ph -> T : NN --> ( 0 [,) +oo ) ) |
| 43 |
42
|
frnd |
|- ( ph -> ran T C_ ( 0 [,) +oo ) ) |
| 44 |
|
icossxr |
|- ( 0 [,) +oo ) C_ RR* |
| 45 |
43 44
|
sstrdi |
|- ( ph -> ran T C_ RR* ) |
| 46 |
|
supxrcl |
|- ( ran T C_ RR* -> sup ( ran T , RR* , < ) e. RR* ) |
| 47 |
45 46
|
syl |
|- ( ph -> sup ( ran T , RR* , < ) e. RR* ) |
| 48 |
6
|
rpred |
|- ( ph -> B e. RR ) |
| 49 |
7 48
|
readdcld |
|- ( ph -> ( sup ( ran S , RR* , < ) + B ) e. RR ) |
| 50 |
49
|
rexrd |
|- ( ph -> ( sup ( ran S , RR* , < ) + B ) e. RR* ) |
| 51 |
|
2fveq3 |
|- ( n = m -> ( 1st ` ( F ` n ) ) = ( 1st ` ( F ` m ) ) ) |
| 52 |
|
oveq2 |
|- ( n = m -> ( 2 ^ n ) = ( 2 ^ m ) ) |
| 53 |
52
|
oveq2d |
|- ( n = m -> ( ( B / 2 ) / ( 2 ^ n ) ) = ( ( B / 2 ) / ( 2 ^ m ) ) ) |
| 54 |
51 53
|
oveq12d |
|- ( n = m -> ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) = ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) ) |
| 55 |
|
2fveq3 |
|- ( n = m -> ( 2nd ` ( F ` n ) ) = ( 2nd ` ( F ` m ) ) ) |
| 56 |
55 53
|
oveq12d |
|- ( n = m -> ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) = ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) ) |
| 57 |
54 56
|
opeq12d |
|- ( n = m -> <. ( ( 1st ` ( F ` n ) ) - ( ( B / 2 ) / ( 2 ^ n ) ) ) , ( ( 2nd ` ( F ` n ) ) + ( ( B / 2 ) / ( 2 ^ n ) ) ) >. = <. ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) >. ) |
| 58 |
|
opex |
|- <. ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) >. e. _V |
| 59 |
57 2 58
|
fvmpt |
|- ( m e. NN -> ( G ` m ) = <. ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) >. ) |
| 60 |
59
|
adantl |
|- ( ( ph /\ m e. NN ) -> ( G ` m ) = <. ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) >. ) |
| 61 |
60
|
fveq2d |
|- ( ( ph /\ m e. NN ) -> ( 1st ` ( G ` m ) ) = ( 1st ` <. ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) >. ) ) |
| 62 |
|
ovex |
|- ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) e. _V |
| 63 |
|
ovex |
|- ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) e. _V |
| 64 |
62 63
|
op1st |
|- ( 1st ` <. ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) >. ) = ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) |
| 65 |
61 64
|
eqtrdi |
|- ( ( ph /\ m e. NN ) -> ( 1st ` ( G ` m ) ) = ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) ) |
| 66 |
|
ovolfcl |
|- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ m e. NN ) -> ( ( 1st ` ( F ` m ) ) e. RR /\ ( 2nd ` ( F ` m ) ) e. RR /\ ( 1st ` ( F ` m ) ) <_ ( 2nd ` ( F ` m ) ) ) ) |
| 67 |
4 66
|
sylan |
|- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( F ` m ) ) e. RR /\ ( 2nd ` ( F ` m ) ) e. RR /\ ( 1st ` ( F ` m ) ) <_ ( 2nd ` ( F ` m ) ) ) ) |
| 68 |
67
|
simp1d |
|- ( ( ph /\ m e. NN ) -> ( 1st ` ( F ` m ) ) e. RR ) |
| 69 |
16
|
adantr |
|- ( ( ph /\ m e. NN ) -> ( B / 2 ) e. RR+ ) |
| 70 |
|
nnnn0 |
|- ( m e. NN -> m e. NN0 ) |
| 71 |
70
|
adantl |
|- ( ( ph /\ m e. NN ) -> m e. NN0 ) |
| 72 |
|
nnexpcl |
|- ( ( 2 e. NN /\ m e. NN0 ) -> ( 2 ^ m ) e. NN ) |
| 73 |
18 71 72
|
sylancr |
|- ( ( ph /\ m e. NN ) -> ( 2 ^ m ) e. NN ) |
| 74 |
73
|
nnrpd |
|- ( ( ph /\ m e. NN ) -> ( 2 ^ m ) e. RR+ ) |
| 75 |
69 74
|
rpdivcld |
|- ( ( ph /\ m e. NN ) -> ( ( B / 2 ) / ( 2 ^ m ) ) e. RR+ ) |
| 76 |
68 75
|
ltsubrpd |
|- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) < ( 1st ` ( F ` m ) ) ) |
| 77 |
65 76
|
eqbrtrd |
|- ( ( ph /\ m e. NN ) -> ( 1st ` ( G ` m ) ) < ( 1st ` ( F ` m ) ) ) |
| 78 |
77
|
adantlr |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> ( 1st ` ( G ` m ) ) < ( 1st ` ( F ` m ) ) ) |
| 79 |
|
ovolfcl |
|- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ m e. NN ) -> ( ( 1st ` ( G ` m ) ) e. RR /\ ( 2nd ` ( G ` m ) ) e. RR /\ ( 1st ` ( G ` m ) ) <_ ( 2nd ` ( G ` m ) ) ) ) |
| 80 |
39 79
|
sylan |
|- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( G ` m ) ) e. RR /\ ( 2nd ` ( G ` m ) ) e. RR /\ ( 1st ` ( G ` m ) ) <_ ( 2nd ` ( G ` m ) ) ) ) |
| 81 |
80
|
simp1d |
|- ( ( ph /\ m e. NN ) -> ( 1st ` ( G ` m ) ) e. RR ) |
| 82 |
81
|
adantlr |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> ( 1st ` ( G ` m ) ) e. RR ) |
| 83 |
68
|
adantlr |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> ( 1st ` ( F ` m ) ) e. RR ) |
| 84 |
10
|
sselda |
|- ( ( ph /\ z e. A ) -> z e. RR ) |
| 85 |
84
|
adantr |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> z e. RR ) |
| 86 |
|
ltletr |
|- ( ( ( 1st ` ( G ` m ) ) e. RR /\ ( 1st ` ( F ` m ) ) e. RR /\ z e. RR ) -> ( ( ( 1st ` ( G ` m ) ) < ( 1st ` ( F ` m ) ) /\ ( 1st ` ( F ` m ) ) <_ z ) -> ( 1st ` ( G ` m ) ) < z ) ) |
| 87 |
82 83 85 86
|
syl3anc |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> ( ( ( 1st ` ( G ` m ) ) < ( 1st ` ( F ` m ) ) /\ ( 1st ` ( F ` m ) ) <_ z ) -> ( 1st ` ( G ` m ) ) < z ) ) |
| 88 |
78 87
|
mpand |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> ( ( 1st ` ( F ` m ) ) <_ z -> ( 1st ` ( G ` m ) ) < z ) ) |
| 89 |
67
|
simp2d |
|- ( ( ph /\ m e. NN ) -> ( 2nd ` ( F ` m ) ) e. RR ) |
| 90 |
89 75
|
ltaddrpd |
|- ( ( ph /\ m e. NN ) -> ( 2nd ` ( F ` m ) ) < ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) ) |
| 91 |
60
|
fveq2d |
|- ( ( ph /\ m e. NN ) -> ( 2nd ` ( G ` m ) ) = ( 2nd ` <. ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) >. ) ) |
| 92 |
62 63
|
op2nd |
|- ( 2nd ` <. ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) , ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) >. ) = ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) |
| 93 |
91 92
|
eqtrdi |
|- ( ( ph /\ m e. NN ) -> ( 2nd ` ( G ` m ) ) = ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) ) |
| 94 |
90 93
|
breqtrrd |
|- ( ( ph /\ m e. NN ) -> ( 2nd ` ( F ` m ) ) < ( 2nd ` ( G ` m ) ) ) |
| 95 |
94
|
adantlr |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> ( 2nd ` ( F ` m ) ) < ( 2nd ` ( G ` m ) ) ) |
| 96 |
89
|
adantlr |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> ( 2nd ` ( F ` m ) ) e. RR ) |
| 97 |
80
|
simp2d |
|- ( ( ph /\ m e. NN ) -> ( 2nd ` ( G ` m ) ) e. RR ) |
| 98 |
97
|
adantlr |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> ( 2nd ` ( G ` m ) ) e. RR ) |
| 99 |
|
lelttr |
|- ( ( z e. RR /\ ( 2nd ` ( F ` m ) ) e. RR /\ ( 2nd ` ( G ` m ) ) e. RR ) -> ( ( z <_ ( 2nd ` ( F ` m ) ) /\ ( 2nd ` ( F ` m ) ) < ( 2nd ` ( G ` m ) ) ) -> z < ( 2nd ` ( G ` m ) ) ) ) |
| 100 |
85 96 98 99
|
syl3anc |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> ( ( z <_ ( 2nd ` ( F ` m ) ) /\ ( 2nd ` ( F ` m ) ) < ( 2nd ` ( G ` m ) ) ) -> z < ( 2nd ` ( G ` m ) ) ) ) |
| 101 |
95 100
|
mpan2d |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> ( z <_ ( 2nd ` ( F ` m ) ) -> z < ( 2nd ` ( G ` m ) ) ) ) |
| 102 |
88 101
|
anim12d |
|- ( ( ( ph /\ z e. A ) /\ m e. NN ) -> ( ( ( 1st ` ( F ` m ) ) <_ z /\ z <_ ( 2nd ` ( F ` m ) ) ) -> ( ( 1st ` ( G ` m ) ) < z /\ z < ( 2nd ` ( G ` m ) ) ) ) ) |
| 103 |
102
|
reximdva |
|- ( ( ph /\ z e. A ) -> ( E. m e. NN ( ( 1st ` ( F ` m ) ) <_ z /\ z <_ ( 2nd ` ( F ` m ) ) ) -> E. m e. NN ( ( 1st ` ( G ` m ) ) < z /\ z < ( 2nd ` ( G ` m ) ) ) ) ) |
| 104 |
103
|
ralimdva |
|- ( ph -> ( A. z e. A E. m e. NN ( ( 1st ` ( F ` m ) ) <_ z /\ z <_ ( 2nd ` ( F ` m ) ) ) -> A. z e. A E. m e. NN ( ( 1st ` ( G ` m ) ) < z /\ z < ( 2nd ` ( G ` m ) ) ) ) ) |
| 105 |
|
ovolficc |
|- ( ( A C_ RR /\ F : NN --> ( <_ i^i ( RR X. RR ) ) ) -> ( A C_ U. ran ( [,] o. F ) <-> A. z e. A E. m e. NN ( ( 1st ` ( F ` m ) ) <_ z /\ z <_ ( 2nd ` ( F ` m ) ) ) ) ) |
| 106 |
10 4 105
|
syl2anc |
|- ( ph -> ( A C_ U. ran ( [,] o. F ) <-> A. z e. A E. m e. NN ( ( 1st ` ( F ` m ) ) <_ z /\ z <_ ( 2nd ` ( F ` m ) ) ) ) ) |
| 107 |
|
ovolfioo |
|- ( ( A C_ RR /\ G : NN --> ( <_ i^i ( RR X. RR ) ) ) -> ( A C_ U. ran ( (,) o. G ) <-> A. z e. A E. m e. NN ( ( 1st ` ( G ` m ) ) < z /\ z < ( 2nd ` ( G ` m ) ) ) ) ) |
| 108 |
10 39 107
|
syl2anc |
|- ( ph -> ( A C_ U. ran ( (,) o. G ) <-> A. z e. A E. m e. NN ( ( 1st ` ( G ` m ) ) < z /\ z < ( 2nd ` ( G ` m ) ) ) ) ) |
| 109 |
104 106 108
|
3imtr4d |
|- ( ph -> ( A C_ U. ran ( [,] o. F ) -> A C_ U. ran ( (,) o. G ) ) ) |
| 110 |
5 109
|
mpd |
|- ( ph -> A C_ U. ran ( (,) o. G ) ) |
| 111 |
3
|
ovollb |
|- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( (,) o. G ) ) -> ( vol* ` A ) <_ sup ( ran T , RR* , < ) ) |
| 112 |
39 110 111
|
syl2anc |
|- ( ph -> ( vol* ` A ) <_ sup ( ran T , RR* , < ) ) |
| 113 |
3
|
fveq1i |
|- ( T ` k ) = ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` k ) |
| 114 |
|
fzfid |
|- ( ( ph /\ k e. NN ) -> ( 1 ... k ) e. Fin ) |
| 115 |
|
rge0ssre |
|- ( 0 [,) +oo ) C_ RR |
| 116 |
|
eqid |
|- ( ( abs o. - ) o. F ) = ( ( abs o. - ) o. F ) |
| 117 |
116
|
ovolfsf |
|- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) ) |
| 118 |
4 117
|
syl |
|- ( ph -> ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) ) |
| 119 |
118
|
adantr |
|- ( ( ph /\ k e. NN ) -> ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) ) |
| 120 |
|
elfznn |
|- ( m e. ( 1 ... k ) -> m e. NN ) |
| 121 |
|
ffvelcdm |
|- ( ( ( ( abs o. - ) o. F ) : NN --> ( 0 [,) +oo ) /\ m e. NN ) -> ( ( ( abs o. - ) o. F ) ` m ) e. ( 0 [,) +oo ) ) |
| 122 |
119 120 121
|
syl2an |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( ( ( abs o. - ) o. F ) ` m ) e. ( 0 [,) +oo ) ) |
| 123 |
115 122
|
sselid |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( ( ( abs o. - ) o. F ) ` m ) e. RR ) |
| 124 |
123
|
recnd |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( ( ( abs o. - ) o. F ) ` m ) e. CC ) |
| 125 |
6
|
adantr |
|- ( ( ph /\ m e. NN ) -> B e. RR+ ) |
| 126 |
125 74
|
rpdivcld |
|- ( ( ph /\ m e. NN ) -> ( B / ( 2 ^ m ) ) e. RR+ ) |
| 127 |
126
|
rpcnd |
|- ( ( ph /\ m e. NN ) -> ( B / ( 2 ^ m ) ) e. CC ) |
| 128 |
120 127
|
sylan2 |
|- ( ( ph /\ m e. ( 1 ... k ) ) -> ( B / ( 2 ^ m ) ) e. CC ) |
| 129 |
128
|
adantlr |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( B / ( 2 ^ m ) ) e. CC ) |
| 130 |
114 124 129
|
fsumadd |
|- ( ( ph /\ k e. NN ) -> sum_ m e. ( 1 ... k ) ( ( ( ( abs o. - ) o. F ) ` m ) + ( B / ( 2 ^ m ) ) ) = ( sum_ m e. ( 1 ... k ) ( ( ( abs o. - ) o. F ) ` m ) + sum_ m e. ( 1 ... k ) ( B / ( 2 ^ m ) ) ) ) |
| 131 |
40
|
ovolfsval |
|- ( ( G : NN --> ( <_ i^i ( RR X. RR ) ) /\ m e. NN ) -> ( ( ( abs o. - ) o. G ) ` m ) = ( ( 2nd ` ( G ` m ) ) - ( 1st ` ( G ` m ) ) ) ) |
| 132 |
39 131
|
sylan |
|- ( ( ph /\ m e. NN ) -> ( ( ( abs o. - ) o. G ) ` m ) = ( ( 2nd ` ( G ` m ) ) - ( 1st ` ( G ` m ) ) ) ) |
| 133 |
89
|
recnd |
|- ( ( ph /\ m e. NN ) -> ( 2nd ` ( F ` m ) ) e. CC ) |
| 134 |
75
|
rpcnd |
|- ( ( ph /\ m e. NN ) -> ( ( B / 2 ) / ( 2 ^ m ) ) e. CC ) |
| 135 |
68
|
recnd |
|- ( ( ph /\ m e. NN ) -> ( 1st ` ( F ` m ) ) e. CC ) |
| 136 |
135 134
|
subcld |
|- ( ( ph /\ m e. NN ) -> ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) e. CC ) |
| 137 |
133 134 136
|
addsubassd |
|- ( ( ph /\ m e. NN ) -> ( ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) - ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) ) = ( ( 2nd ` ( F ` m ) ) + ( ( ( B / 2 ) / ( 2 ^ m ) ) - ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) ) ) ) |
| 138 |
93 65
|
oveq12d |
|- ( ( ph /\ m e. NN ) -> ( ( 2nd ` ( G ` m ) ) - ( 1st ` ( G ` m ) ) ) = ( ( ( 2nd ` ( F ` m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) - ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) ) ) |
| 139 |
133 135 127
|
subadd23d |
|- ( ( ph /\ m e. NN ) -> ( ( ( 2nd ` ( F ` m ) ) - ( 1st ` ( F ` m ) ) ) + ( B / ( 2 ^ m ) ) ) = ( ( 2nd ` ( F ` m ) ) + ( ( B / ( 2 ^ m ) ) - ( 1st ` ( F ` m ) ) ) ) ) |
| 140 |
116
|
ovolfsval |
|- ( ( F : NN --> ( <_ i^i ( RR X. RR ) ) /\ m e. NN ) -> ( ( ( abs o. - ) o. F ) ` m ) = ( ( 2nd ` ( F ` m ) ) - ( 1st ` ( F ` m ) ) ) ) |
| 141 |
4 140
|
sylan |
|- ( ( ph /\ m e. NN ) -> ( ( ( abs o. - ) o. F ) ` m ) = ( ( 2nd ` ( F ` m ) ) - ( 1st ` ( F ` m ) ) ) ) |
| 142 |
141
|
oveq1d |
|- ( ( ph /\ m e. NN ) -> ( ( ( ( abs o. - ) o. F ) ` m ) + ( B / ( 2 ^ m ) ) ) = ( ( ( 2nd ` ( F ` m ) ) - ( 1st ` ( F ` m ) ) ) + ( B / ( 2 ^ m ) ) ) ) |
| 143 |
134 135 134
|
subsub3d |
|- ( ( ph /\ m e. NN ) -> ( ( ( B / 2 ) / ( 2 ^ m ) ) - ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) ) = ( ( ( ( B / 2 ) / ( 2 ^ m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) - ( 1st ` ( F ` m ) ) ) ) |
| 144 |
69
|
rpcnd |
|- ( ( ph /\ m e. NN ) -> ( B / 2 ) e. CC ) |
| 145 |
73
|
nncnd |
|- ( ( ph /\ m e. NN ) -> ( 2 ^ m ) e. CC ) |
| 146 |
73
|
nnne0d |
|- ( ( ph /\ m e. NN ) -> ( 2 ^ m ) =/= 0 ) |
| 147 |
144 144 145 146
|
divdird |
|- ( ( ph /\ m e. NN ) -> ( ( ( B / 2 ) + ( B / 2 ) ) / ( 2 ^ m ) ) = ( ( ( B / 2 ) / ( 2 ^ m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) ) |
| 148 |
125
|
rpcnd |
|- ( ( ph /\ m e. NN ) -> B e. CC ) |
| 149 |
148
|
2halvesd |
|- ( ( ph /\ m e. NN ) -> ( ( B / 2 ) + ( B / 2 ) ) = B ) |
| 150 |
149
|
oveq1d |
|- ( ( ph /\ m e. NN ) -> ( ( ( B / 2 ) + ( B / 2 ) ) / ( 2 ^ m ) ) = ( B / ( 2 ^ m ) ) ) |
| 151 |
147 150
|
eqtr3d |
|- ( ( ph /\ m e. NN ) -> ( ( ( B / 2 ) / ( 2 ^ m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) = ( B / ( 2 ^ m ) ) ) |
| 152 |
151
|
oveq1d |
|- ( ( ph /\ m e. NN ) -> ( ( ( ( B / 2 ) / ( 2 ^ m ) ) + ( ( B / 2 ) / ( 2 ^ m ) ) ) - ( 1st ` ( F ` m ) ) ) = ( ( B / ( 2 ^ m ) ) - ( 1st ` ( F ` m ) ) ) ) |
| 153 |
143 152
|
eqtrd |
|- ( ( ph /\ m e. NN ) -> ( ( ( B / 2 ) / ( 2 ^ m ) ) - ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) ) = ( ( B / ( 2 ^ m ) ) - ( 1st ` ( F ` m ) ) ) ) |
| 154 |
153
|
oveq2d |
|- ( ( ph /\ m e. NN ) -> ( ( 2nd ` ( F ` m ) ) + ( ( ( B / 2 ) / ( 2 ^ m ) ) - ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) ) ) = ( ( 2nd ` ( F ` m ) ) + ( ( B / ( 2 ^ m ) ) - ( 1st ` ( F ` m ) ) ) ) ) |
| 155 |
139 142 154
|
3eqtr4d |
|- ( ( ph /\ m e. NN ) -> ( ( ( ( abs o. - ) o. F ) ` m ) + ( B / ( 2 ^ m ) ) ) = ( ( 2nd ` ( F ` m ) ) + ( ( ( B / 2 ) / ( 2 ^ m ) ) - ( ( 1st ` ( F ` m ) ) - ( ( B / 2 ) / ( 2 ^ m ) ) ) ) ) ) |
| 156 |
137 138 155
|
3eqtr4d |
|- ( ( ph /\ m e. NN ) -> ( ( 2nd ` ( G ` m ) ) - ( 1st ` ( G ` m ) ) ) = ( ( ( ( abs o. - ) o. F ) ` m ) + ( B / ( 2 ^ m ) ) ) ) |
| 157 |
132 156
|
eqtrd |
|- ( ( ph /\ m e. NN ) -> ( ( ( abs o. - ) o. G ) ` m ) = ( ( ( ( abs o. - ) o. F ) ` m ) + ( B / ( 2 ^ m ) ) ) ) |
| 158 |
120 157
|
sylan2 |
|- ( ( ph /\ m e. ( 1 ... k ) ) -> ( ( ( abs o. - ) o. G ) ` m ) = ( ( ( ( abs o. - ) o. F ) ` m ) + ( B / ( 2 ^ m ) ) ) ) |
| 159 |
158
|
adantlr |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( ( ( abs o. - ) o. G ) ` m ) = ( ( ( ( abs o. - ) o. F ) ` m ) + ( B / ( 2 ^ m ) ) ) ) |
| 160 |
|
simpr |
|- ( ( ph /\ k e. NN ) -> k e. NN ) |
| 161 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
| 162 |
160 161
|
eleqtrdi |
|- ( ( ph /\ k e. NN ) -> k e. ( ZZ>= ` 1 ) ) |
| 163 |
124 129
|
addcld |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( ( ( ( abs o. - ) o. F ) ` m ) + ( B / ( 2 ^ m ) ) ) e. CC ) |
| 164 |
159 162 163
|
fsumser |
|- ( ( ph /\ k e. NN ) -> sum_ m e. ( 1 ... k ) ( ( ( ( abs o. - ) o. F ) ` m ) + ( B / ( 2 ^ m ) ) ) = ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` k ) ) |
| 165 |
|
eqidd |
|- ( ( ( ph /\ k e. NN ) /\ m e. ( 1 ... k ) ) -> ( ( ( abs o. - ) o. F ) ` m ) = ( ( ( abs o. - ) o. F ) ` m ) ) |
| 166 |
165 162 124
|
fsumser |
|- ( ( ph /\ k e. NN ) -> sum_ m e. ( 1 ... k ) ( ( ( abs o. - ) o. F ) ` m ) = ( seq 1 ( + , ( ( abs o. - ) o. F ) ) ` k ) ) |
| 167 |
1
|
fveq1i |
|- ( S ` k ) = ( seq 1 ( + , ( ( abs o. - ) o. F ) ) ` k ) |
| 168 |
166 167
|
eqtr4di |
|- ( ( ph /\ k e. NN ) -> sum_ m e. ( 1 ... k ) ( ( ( abs o. - ) o. F ) ` m ) = ( S ` k ) ) |
| 169 |
6
|
adantr |
|- ( ( ph /\ k e. NN ) -> B e. RR+ ) |
| 170 |
169
|
rpcnd |
|- ( ( ph /\ k e. NN ) -> B e. CC ) |
| 171 |
|
geo2sum |
|- ( ( k e. NN /\ B e. CC ) -> sum_ m e. ( 1 ... k ) ( B / ( 2 ^ m ) ) = ( B - ( B / ( 2 ^ k ) ) ) ) |
| 172 |
160 170 171
|
syl2anc |
|- ( ( ph /\ k e. NN ) -> sum_ m e. ( 1 ... k ) ( B / ( 2 ^ m ) ) = ( B - ( B / ( 2 ^ k ) ) ) ) |
| 173 |
168 172
|
oveq12d |
|- ( ( ph /\ k e. NN ) -> ( sum_ m e. ( 1 ... k ) ( ( ( abs o. - ) o. F ) ` m ) + sum_ m e. ( 1 ... k ) ( B / ( 2 ^ m ) ) ) = ( ( S ` k ) + ( B - ( B / ( 2 ^ k ) ) ) ) ) |
| 174 |
130 164 173
|
3eqtr3d |
|- ( ( ph /\ k e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. G ) ) ` k ) = ( ( S ` k ) + ( B - ( B / ( 2 ^ k ) ) ) ) ) |
| 175 |
113 174
|
eqtrid |
|- ( ( ph /\ k e. NN ) -> ( T ` k ) = ( ( S ` k ) + ( B - ( B / ( 2 ^ k ) ) ) ) ) |
| 176 |
116 1
|
ovolsf |
|- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> S : NN --> ( 0 [,) +oo ) ) |
| 177 |
4 176
|
syl |
|- ( ph -> S : NN --> ( 0 [,) +oo ) ) |
| 178 |
177
|
ffvelcdmda |
|- ( ( ph /\ k e. NN ) -> ( S ` k ) e. ( 0 [,) +oo ) ) |
| 179 |
115 178
|
sselid |
|- ( ( ph /\ k e. NN ) -> ( S ` k ) e. RR ) |
| 180 |
169
|
rpred |
|- ( ( ph /\ k e. NN ) -> B e. RR ) |
| 181 |
|
nnnn0 |
|- ( k e. NN -> k e. NN0 ) |
| 182 |
181
|
adantl |
|- ( ( ph /\ k e. NN ) -> k e. NN0 ) |
| 183 |
|
nnexpcl |
|- ( ( 2 e. NN /\ k e. NN0 ) -> ( 2 ^ k ) e. NN ) |
| 184 |
18 182 183
|
sylancr |
|- ( ( ph /\ k e. NN ) -> ( 2 ^ k ) e. NN ) |
| 185 |
184
|
nnrpd |
|- ( ( ph /\ k e. NN ) -> ( 2 ^ k ) e. RR+ ) |
| 186 |
169 185
|
rpdivcld |
|- ( ( ph /\ k e. NN ) -> ( B / ( 2 ^ k ) ) e. RR+ ) |
| 187 |
186
|
rpred |
|- ( ( ph /\ k e. NN ) -> ( B / ( 2 ^ k ) ) e. RR ) |
| 188 |
180 187
|
resubcld |
|- ( ( ph /\ k e. NN ) -> ( B - ( B / ( 2 ^ k ) ) ) e. RR ) |
| 189 |
7
|
adantr |
|- ( ( ph /\ k e. NN ) -> sup ( ran S , RR* , < ) e. RR ) |
| 190 |
177
|
frnd |
|- ( ph -> ran S C_ ( 0 [,) +oo ) ) |
| 191 |
190 44
|
sstrdi |
|- ( ph -> ran S C_ RR* ) |
| 192 |
191
|
adantr |
|- ( ( ph /\ k e. NN ) -> ran S C_ RR* ) |
| 193 |
177
|
ffnd |
|- ( ph -> S Fn NN ) |
| 194 |
|
fnfvelrn |
|- ( ( S Fn NN /\ k e. NN ) -> ( S ` k ) e. ran S ) |
| 195 |
193 194
|
sylan |
|- ( ( ph /\ k e. NN ) -> ( S ` k ) e. ran S ) |
| 196 |
|
supxrub |
|- ( ( ran S C_ RR* /\ ( S ` k ) e. ran S ) -> ( S ` k ) <_ sup ( ran S , RR* , < ) ) |
| 197 |
192 195 196
|
syl2anc |
|- ( ( ph /\ k e. NN ) -> ( S ` k ) <_ sup ( ran S , RR* , < ) ) |
| 198 |
180 186
|
ltsubrpd |
|- ( ( ph /\ k e. NN ) -> ( B - ( B / ( 2 ^ k ) ) ) < B ) |
| 199 |
188 180 198
|
ltled |
|- ( ( ph /\ k e. NN ) -> ( B - ( B / ( 2 ^ k ) ) ) <_ B ) |
| 200 |
179 188 189 180 197 199
|
le2addd |
|- ( ( ph /\ k e. NN ) -> ( ( S ` k ) + ( B - ( B / ( 2 ^ k ) ) ) ) <_ ( sup ( ran S , RR* , < ) + B ) ) |
| 201 |
175 200
|
eqbrtrd |
|- ( ( ph /\ k e. NN ) -> ( T ` k ) <_ ( sup ( ran S , RR* , < ) + B ) ) |
| 202 |
201
|
ralrimiva |
|- ( ph -> A. k e. NN ( T ` k ) <_ ( sup ( ran S , RR* , < ) + B ) ) |
| 203 |
|
ffn |
|- ( T : NN --> ( 0 [,) +oo ) -> T Fn NN ) |
| 204 |
|
breq1 |
|- ( y = ( T ` k ) -> ( y <_ ( sup ( ran S , RR* , < ) + B ) <-> ( T ` k ) <_ ( sup ( ran S , RR* , < ) + B ) ) ) |
| 205 |
204
|
ralrn |
|- ( T Fn NN -> ( A. y e. ran T y <_ ( sup ( ran S , RR* , < ) + B ) <-> A. k e. NN ( T ` k ) <_ ( sup ( ran S , RR* , < ) + B ) ) ) |
| 206 |
42 203 205
|
3syl |
|- ( ph -> ( A. y e. ran T y <_ ( sup ( ran S , RR* , < ) + B ) <-> A. k e. NN ( T ` k ) <_ ( sup ( ran S , RR* , < ) + B ) ) ) |
| 207 |
202 206
|
mpbird |
|- ( ph -> A. y e. ran T y <_ ( sup ( ran S , RR* , < ) + B ) ) |
| 208 |
|
supxrleub |
|- ( ( ran T C_ RR* /\ ( sup ( ran S , RR* , < ) + B ) e. RR* ) -> ( sup ( ran T , RR* , < ) <_ ( sup ( ran S , RR* , < ) + B ) <-> A. y e. ran T y <_ ( sup ( ran S , RR* , < ) + B ) ) ) |
| 209 |
45 50 208
|
syl2anc |
|- ( ph -> ( sup ( ran T , RR* , < ) <_ ( sup ( ran S , RR* , < ) + B ) <-> A. y e. ran T y <_ ( sup ( ran S , RR* , < ) + B ) ) ) |
| 210 |
207 209
|
mpbird |
|- ( ph -> sup ( ran T , RR* , < ) <_ ( sup ( ran S , RR* , < ) + B ) ) |
| 211 |
12 47 50 112 210
|
xrletrd |
|- ( ph -> ( vol* ` A ) <_ ( sup ( ran S , RR* , < ) + B ) ) |