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 |
|
ffvelrn |
|- ( ( ( ( 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
|
ffvelrnda |
|- ( ( 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 ) ) |