Step |
Hyp |
Ref |
Expression |
1 |
|
vonicc.x |
|- ( ph -> X e. Fin ) |
2 |
|
vonicc.a |
|- ( ph -> A : X --> RR ) |
3 |
|
vonicc.b |
|- ( ph -> B : X --> RR ) |
4 |
|
vonicc.i |
|- I = X_ k e. X ( ( A ` k ) [,] ( B ` k ) ) |
5 |
|
vonicc.l |
|- L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) ) |
6 |
2
|
adantr |
|- ( ( ph /\ X = (/) ) -> A : X --> RR ) |
7 |
|
feq2 |
|- ( X = (/) -> ( A : X --> RR <-> A : (/) --> RR ) ) |
8 |
7
|
adantl |
|- ( ( ph /\ X = (/) ) -> ( A : X --> RR <-> A : (/) --> RR ) ) |
9 |
6 8
|
mpbid |
|- ( ( ph /\ X = (/) ) -> A : (/) --> RR ) |
10 |
3
|
adantr |
|- ( ( ph /\ X = (/) ) -> B : X --> RR ) |
11 |
|
feq2 |
|- ( X = (/) -> ( B : X --> RR <-> B : (/) --> RR ) ) |
12 |
11
|
adantl |
|- ( ( ph /\ X = (/) ) -> ( B : X --> RR <-> B : (/) --> RR ) ) |
13 |
10 12
|
mpbid |
|- ( ( ph /\ X = (/) ) -> B : (/) --> RR ) |
14 |
5 9 13
|
hoidmv0val |
|- ( ( ph /\ X = (/) ) -> ( A ( L ` (/) ) B ) = 0 ) |
15 |
14
|
eqcomd |
|- ( ( ph /\ X = (/) ) -> 0 = ( A ( L ` (/) ) B ) ) |
16 |
|
fveq2 |
|- ( X = (/) -> ( voln ` X ) = ( voln ` (/) ) ) |
17 |
4
|
a1i |
|- ( X = (/) -> I = X_ k e. X ( ( A ` k ) [,] ( B ` k ) ) ) |
18 |
|
ixpeq1 |
|- ( X = (/) -> X_ k e. X ( ( A ` k ) [,] ( B ` k ) ) = X_ k e. (/) ( ( A ` k ) [,] ( B ` k ) ) ) |
19 |
17 18
|
eqtrd |
|- ( X = (/) -> I = X_ k e. (/) ( ( A ` k ) [,] ( B ` k ) ) ) |
20 |
16 19
|
fveq12d |
|- ( X = (/) -> ( ( voln ` X ) ` I ) = ( ( voln ` (/) ) ` X_ k e. (/) ( ( A ` k ) [,] ( B ` k ) ) ) ) |
21 |
20
|
adantl |
|- ( ( ph /\ X = (/) ) -> ( ( voln ` X ) ` I ) = ( ( voln ` (/) ) ` X_ k e. (/) ( ( A ` k ) [,] ( B ` k ) ) ) ) |
22 |
|
0fin |
|- (/) e. Fin |
23 |
22
|
a1i |
|- ( ( ph /\ X = (/) ) -> (/) e. Fin ) |
24 |
|
eqid |
|- dom ( voln ` (/) ) = dom ( voln ` (/) ) |
25 |
23 24 9 13
|
iccvonmbl |
|- ( ( ph /\ X = (/) ) -> X_ k e. (/) ( ( A ` k ) [,] ( B ` k ) ) e. dom ( voln ` (/) ) ) |
26 |
25
|
von0val |
|- ( ( ph /\ X = (/) ) -> ( ( voln ` (/) ) ` X_ k e. (/) ( ( A ` k ) [,] ( B ` k ) ) ) = 0 ) |
27 |
21 26
|
eqtrd |
|- ( ( ph /\ X = (/) ) -> ( ( voln ` X ) ` I ) = 0 ) |
28 |
|
fveq2 |
|- ( X = (/) -> ( L ` X ) = ( L ` (/) ) ) |
29 |
28
|
oveqd |
|- ( X = (/) -> ( A ( L ` X ) B ) = ( A ( L ` (/) ) B ) ) |
30 |
29
|
adantl |
|- ( ( ph /\ X = (/) ) -> ( A ( L ` X ) B ) = ( A ( L ` (/) ) B ) ) |
31 |
15 27 30
|
3eqtr4d |
|- ( ( ph /\ X = (/) ) -> ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) ) |
32 |
|
neqne |
|- ( -. X = (/) -> X =/= (/) ) |
33 |
32
|
adantl |
|- ( ( ph /\ -. X = (/) ) -> X =/= (/) ) |
34 |
|
nfv |
|- F/ k ( ph /\ X =/= (/) ) |
35 |
|
nfra1 |
|- F/ k A. k e. X ( A ` k ) <_ ( B ` k ) |
36 |
34 35
|
nfan |
|- F/ k ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) |
37 |
2
|
ffvelrnda |
|- ( ( ph /\ k e. X ) -> ( A ` k ) e. RR ) |
38 |
3
|
ffvelrnda |
|- ( ( ph /\ k e. X ) -> ( B ` k ) e. RR ) |
39 |
|
volico2 |
|- ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = if ( ( A ` k ) <_ ( B ` k ) , ( ( B ` k ) - ( A ` k ) ) , 0 ) ) |
40 |
37 38 39
|
syl2anc |
|- ( ( ph /\ k e. X ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = if ( ( A ` k ) <_ ( B ` k ) , ( ( B ` k ) - ( A ` k ) ) , 0 ) ) |
41 |
40
|
ad4ant14 |
|- ( ( ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) /\ k e. X ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = if ( ( A ` k ) <_ ( B ` k ) , ( ( B ` k ) - ( A ` k ) ) , 0 ) ) |
42 |
|
rspa |
|- ( ( A. k e. X ( A ` k ) <_ ( B ` k ) /\ k e. X ) -> ( A ` k ) <_ ( B ` k ) ) |
43 |
42
|
iftrued |
|- ( ( A. k e. X ( A ` k ) <_ ( B ` k ) /\ k e. X ) -> if ( ( A ` k ) <_ ( B ` k ) , ( ( B ` k ) - ( A ` k ) ) , 0 ) = ( ( B ` k ) - ( A ` k ) ) ) |
44 |
43
|
adantll |
|- ( ( ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) /\ k e. X ) -> if ( ( A ` k ) <_ ( B ` k ) , ( ( B ` k ) - ( A ` k ) ) , 0 ) = ( ( B ` k ) - ( A ` k ) ) ) |
45 |
41 44
|
eqtrd |
|- ( ( ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) /\ k e. X ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = ( ( B ` k ) - ( A ` k ) ) ) |
46 |
45
|
ex |
|- ( ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) -> ( k e. X -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = ( ( B ` k ) - ( A ` k ) ) ) ) |
47 |
36 46
|
ralrimi |
|- ( ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) -> A. k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = ( ( B ` k ) - ( A ` k ) ) ) |
48 |
47
|
prodeq2d |
|- ( ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = prod_ k e. X ( ( B ` k ) - ( A ` k ) ) ) |
49 |
48
|
eqcomd |
|- ( ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) -> prod_ k e. X ( ( B ` k ) - ( A ` k ) ) = prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) |
50 |
|
fveq2 |
|- ( k = j -> ( A ` k ) = ( A ` j ) ) |
51 |
|
fveq2 |
|- ( k = j -> ( B ` k ) = ( B ` j ) ) |
52 |
50 51
|
breq12d |
|- ( k = j -> ( ( A ` k ) <_ ( B ` k ) <-> ( A ` j ) <_ ( B ` j ) ) ) |
53 |
52
|
cbvralvw |
|- ( A. k e. X ( A ` k ) <_ ( B ` k ) <-> A. j e. X ( A ` j ) <_ ( B ` j ) ) |
54 |
53
|
biimpi |
|- ( A. k e. X ( A ` k ) <_ ( B ` k ) -> A. j e. X ( A ` j ) <_ ( B ` j ) ) |
55 |
54
|
adantl |
|- ( ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) -> A. j e. X ( A ` j ) <_ ( B ` j ) ) |
56 |
1
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> X e. Fin ) |
57 |
56
|
adantr |
|- ( ( ( ph /\ X =/= (/) ) /\ A. j e. X ( A ` j ) <_ ( B ` j ) ) -> X e. Fin ) |
58 |
2
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> A : X --> RR ) |
59 |
58
|
adantr |
|- ( ( ( ph /\ X =/= (/) ) /\ A. j e. X ( A ` j ) <_ ( B ` j ) ) -> A : X --> RR ) |
60 |
3
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> B : X --> RR ) |
61 |
60
|
adantr |
|- ( ( ( ph /\ X =/= (/) ) /\ A. j e. X ( A ` j ) <_ ( B ` j ) ) -> B : X --> RR ) |
62 |
|
simpr |
|- ( ( ph /\ X =/= (/) ) -> X =/= (/) ) |
63 |
62
|
adantr |
|- ( ( ( ph /\ X =/= (/) ) /\ A. j e. X ( A ` j ) <_ ( B ` j ) ) -> X =/= (/) ) |
64 |
53 42
|
sylanbr |
|- ( ( A. j e. X ( A ` j ) <_ ( B ` j ) /\ k e. X ) -> ( A ` k ) <_ ( B ` k ) ) |
65 |
64
|
adantll |
|- ( ( ( ( ph /\ X =/= (/) ) /\ A. j e. X ( A ` j ) <_ ( B ` j ) ) /\ k e. X ) -> ( A ` k ) <_ ( B ` k ) ) |
66 |
|
fveq2 |
|- ( j = k -> ( B ` j ) = ( B ` k ) ) |
67 |
66
|
oveq1d |
|- ( j = k -> ( ( B ` j ) + ( 1 / m ) ) = ( ( B ` k ) + ( 1 / m ) ) ) |
68 |
67
|
cbvmptv |
|- ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) = ( k e. X |-> ( ( B ` k ) + ( 1 / m ) ) ) |
69 |
68
|
mpteq2i |
|- ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) = ( m e. NN |-> ( k e. X |-> ( ( B ` k ) + ( 1 / m ) ) ) ) |
70 |
|
oveq2 |
|- ( m = n -> ( 1 / m ) = ( 1 / n ) ) |
71 |
70
|
oveq2d |
|- ( m = n -> ( ( B ` k ) + ( 1 / m ) ) = ( ( B ` k ) + ( 1 / n ) ) ) |
72 |
71
|
mpteq2dv |
|- ( m = n -> ( k e. X |-> ( ( B ` k ) + ( 1 / m ) ) ) = ( k e. X |-> ( ( B ` k ) + ( 1 / n ) ) ) ) |
73 |
72
|
cbvmptv |
|- ( m e. NN |-> ( k e. X |-> ( ( B ` k ) + ( 1 / m ) ) ) ) = ( n e. NN |-> ( k e. X |-> ( ( B ` k ) + ( 1 / n ) ) ) ) |
74 |
69 73
|
eqtri |
|- ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) = ( n e. NN |-> ( k e. X |-> ( ( B ` k ) + ( 1 / n ) ) ) ) |
75 |
|
fveq2 |
|- ( i = n -> ( ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) ` i ) = ( ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) ` n ) ) |
76 |
75
|
fveq1d |
|- ( i = n -> ( ( ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) ` i ) ` k ) = ( ( ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) ` n ) ` k ) ) |
77 |
76
|
oveq2d |
|- ( i = n -> ( ( A ` k ) [,) ( ( ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) ` i ) ` k ) ) = ( ( A ` k ) [,) ( ( ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) ` n ) ` k ) ) ) |
78 |
77
|
ixpeq2dv |
|- ( i = n -> X_ k e. X ( ( A ` k ) [,) ( ( ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) ` i ) ` k ) ) = X_ k e. X ( ( A ` k ) [,) ( ( ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) ` n ) ` k ) ) ) |
79 |
78
|
cbvmptv |
|- ( i e. NN |-> X_ k e. X ( ( A ` k ) [,) ( ( ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) ` i ) ` k ) ) ) = ( n e. NN |-> X_ k e. X ( ( A ` k ) [,) ( ( ( m e. NN |-> ( j e. X |-> ( ( B ` j ) + ( 1 / m ) ) ) ) ` n ) ` k ) ) ) |
80 |
57 59 61 63 65 4 74 79
|
vonicclem2 |
|- ( ( ( ph /\ X =/= (/) ) /\ A. j e. X ( A ` j ) <_ ( B ` j ) ) -> ( ( voln ` X ) ` I ) = prod_ k e. X ( ( B ` k ) - ( A ` k ) ) ) |
81 |
55 80
|
syldan |
|- ( ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) -> ( ( voln ` X ) ` I ) = prod_ k e. X ( ( B ` k ) - ( A ` k ) ) ) |
82 |
5 56 62 58 60
|
hoidmvn0val |
|- ( ( ph /\ X =/= (/) ) -> ( A ( L ` X ) B ) = prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) |
83 |
82
|
adantr |
|- ( ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) -> ( A ( L ` X ) B ) = prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) |
84 |
49 81 83
|
3eqtr4d |
|- ( ( ( ph /\ X =/= (/) ) /\ A. k e. X ( A ` k ) <_ ( B ` k ) ) -> ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) ) |
85 |
|
rexnal |
|- ( E. k e. X -. ( A ` k ) <_ ( B ` k ) <-> -. A. k e. X ( A ` k ) <_ ( B ` k ) ) |
86 |
85
|
bicomi |
|- ( -. A. k e. X ( A ` k ) <_ ( B ` k ) <-> E. k e. X -. ( A ` k ) <_ ( B ` k ) ) |
87 |
86
|
biimpi |
|- ( -. A. k e. X ( A ` k ) <_ ( B ` k ) -> E. k e. X -. ( A ` k ) <_ ( B ` k ) ) |
88 |
87
|
adantl |
|- ( ( ph /\ -. A. k e. X ( A ` k ) <_ ( B ` k ) ) -> E. k e. X -. ( A ` k ) <_ ( B ` k ) ) |
89 |
|
simpr |
|- ( ( ( ph /\ k e. X ) /\ -. ( A ` k ) <_ ( B ` k ) ) -> -. ( A ` k ) <_ ( B ` k ) ) |
90 |
38
|
adantr |
|- ( ( ( ph /\ k e. X ) /\ -. ( A ` k ) <_ ( B ` k ) ) -> ( B ` k ) e. RR ) |
91 |
37
|
adantr |
|- ( ( ( ph /\ k e. X ) /\ -. ( A ` k ) <_ ( B ` k ) ) -> ( A ` k ) e. RR ) |
92 |
90 91
|
ltnled |
|- ( ( ( ph /\ k e. X ) /\ -. ( A ` k ) <_ ( B ` k ) ) -> ( ( B ` k ) < ( A ` k ) <-> -. ( A ` k ) <_ ( B ` k ) ) ) |
93 |
89 92
|
mpbird |
|- ( ( ( ph /\ k e. X ) /\ -. ( A ` k ) <_ ( B ` k ) ) -> ( B ` k ) < ( A ` k ) ) |
94 |
93
|
ex |
|- ( ( ph /\ k e. X ) -> ( -. ( A ` k ) <_ ( B ` k ) -> ( B ` k ) < ( A ` k ) ) ) |
95 |
94
|
reximdva |
|- ( ph -> ( E. k e. X -. ( A ` k ) <_ ( B ` k ) -> E. k e. X ( B ` k ) < ( A ` k ) ) ) |
96 |
95
|
adantr |
|- ( ( ph /\ -. A. k e. X ( A ` k ) <_ ( B ` k ) ) -> ( E. k e. X -. ( A ` k ) <_ ( B ` k ) -> E. k e. X ( B ` k ) < ( A ` k ) ) ) |
97 |
88 96
|
mpd |
|- ( ( ph /\ -. A. k e. X ( A ` k ) <_ ( B ` k ) ) -> E. k e. X ( B ` k ) < ( A ` k ) ) |
98 |
97
|
adantlr |
|- ( ( ( ph /\ X =/= (/) ) /\ -. A. k e. X ( A ` k ) <_ ( B ` k ) ) -> E. k e. X ( B ` k ) < ( A ` k ) ) |
99 |
|
nfcv |
|- F/_ k ( voln ` X ) |
100 |
|
nfixp1 |
|- F/_ k X_ k e. X ( ( A ` k ) [,] ( B ` k ) ) |
101 |
4 100
|
nfcxfr |
|- F/_ k I |
102 |
99 101
|
nffv |
|- F/_ k ( ( voln ` X ) ` I ) |
103 |
|
nfcv |
|- F/_ k A |
104 |
|
nfcv |
|- F/_ k Fin |
105 |
|
nfcv |
|- F/_ k ( RR ^m x ) |
106 |
|
nfv |
|- F/ k x = (/) |
107 |
|
nfcv |
|- F/_ k 0 |
108 |
|
nfcv |
|- F/_ k x |
109 |
108
|
nfcprod1 |
|- F/_ k prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) |
110 |
106 107 109
|
nfif |
|- F/_ k if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) |
111 |
105 105 110
|
nfmpo |
|- F/_ k ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) |
112 |
104 111
|
nfmpt |
|- F/_ k ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) ) |
113 |
5 112
|
nfcxfr |
|- F/_ k L |
114 |
|
nfcv |
|- F/_ k X |
115 |
113 114
|
nffv |
|- F/_ k ( L ` X ) |
116 |
|
nfcv |
|- F/_ k B |
117 |
103 115 116
|
nfov |
|- F/_ k ( A ( L ` X ) B ) |
118 |
102 117
|
nfeq |
|- F/ k ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) |
119 |
1
|
vonmea |
|- ( ph -> ( voln ` X ) e. Meas ) |
120 |
119
|
mea0 |
|- ( ph -> ( ( voln ` X ) ` (/) ) = 0 ) |
121 |
120
|
3ad2ant1 |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> ( ( voln ` X ) ` (/) ) = 0 ) |
122 |
4
|
a1i |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> I = X_ k e. X ( ( A ` k ) [,] ( B ` k ) ) ) |
123 |
|
simp2 |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> k e. X ) |
124 |
|
simp3 |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> ( B ` k ) < ( A ` k ) ) |
125 |
|
ressxr |
|- RR C_ RR* |
126 |
125 37
|
sselid |
|- ( ( ph /\ k e. X ) -> ( A ` k ) e. RR* ) |
127 |
125 38
|
sselid |
|- ( ( ph /\ k e. X ) -> ( B ` k ) e. RR* ) |
128 |
|
icc0 |
|- ( ( ( A ` k ) e. RR* /\ ( B ` k ) e. RR* ) -> ( ( ( A ` k ) [,] ( B ` k ) ) = (/) <-> ( B ` k ) < ( A ` k ) ) ) |
129 |
126 127 128
|
syl2anc |
|- ( ( ph /\ k e. X ) -> ( ( ( A ` k ) [,] ( B ` k ) ) = (/) <-> ( B ` k ) < ( A ` k ) ) ) |
130 |
129
|
3adant3 |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> ( ( ( A ` k ) [,] ( B ` k ) ) = (/) <-> ( B ` k ) < ( A ` k ) ) ) |
131 |
124 130
|
mpbird |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> ( ( A ` k ) [,] ( B ` k ) ) = (/) ) |
132 |
|
rspe |
|- ( ( k e. X /\ ( ( A ` k ) [,] ( B ` k ) ) = (/) ) -> E. k e. X ( ( A ` k ) [,] ( B ` k ) ) = (/) ) |
133 |
123 131 132
|
syl2anc |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> E. k e. X ( ( A ` k ) [,] ( B ` k ) ) = (/) ) |
134 |
|
ixp0 |
|- ( E. k e. X ( ( A ` k ) [,] ( B ` k ) ) = (/) -> X_ k e. X ( ( A ` k ) [,] ( B ` k ) ) = (/) ) |
135 |
133 134
|
syl |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> X_ k e. X ( ( A ` k ) [,] ( B ` k ) ) = (/) ) |
136 |
122 135
|
eqtrd |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> I = (/) ) |
137 |
136
|
fveq2d |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> ( ( voln ` X ) ` I ) = ( ( voln ` X ) ` (/) ) ) |
138 |
|
ne0i |
|- ( k e. X -> X =/= (/) ) |
139 |
138
|
adantl |
|- ( ( ph /\ k e. X ) -> X =/= (/) ) |
140 |
139 82
|
syldan |
|- ( ( ph /\ k e. X ) -> ( A ( L ` X ) B ) = prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) |
141 |
140
|
3adant3 |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> ( A ( L ` X ) B ) = prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) |
142 |
|
eleq1w |
|- ( j = k -> ( j e. X <-> k e. X ) ) |
143 |
|
fveq2 |
|- ( j = k -> ( A ` j ) = ( A ` k ) ) |
144 |
66 143
|
breq12d |
|- ( j = k -> ( ( B ` j ) < ( A ` j ) <-> ( B ` k ) < ( A ` k ) ) ) |
145 |
142 144
|
3anbi23d |
|- ( j = k -> ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) <-> ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) ) ) |
146 |
145
|
imbi1d |
|- ( j = k -> ( ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = 0 ) <-> ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = 0 ) ) ) |
147 |
|
nfv |
|- F/ k ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) |
148 |
1
|
3ad2ant1 |
|- ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) -> X e. Fin ) |
149 |
|
volicore |
|- ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) e. RR ) |
150 |
37 38 149
|
syl2anc |
|- ( ( ph /\ k e. X ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) e. RR ) |
151 |
150
|
recnd |
|- ( ( ph /\ k e. X ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) e. CC ) |
152 |
151
|
3ad2antl1 |
|- ( ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) /\ k e. X ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) e. CC ) |
153 |
|
simp2 |
|- ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) -> j e. X ) |
154 |
50 51
|
oveq12d |
|- ( k = j -> ( ( A ` k ) [,) ( B ` k ) ) = ( ( A ` j ) [,) ( B ` j ) ) ) |
155 |
154
|
fveq2d |
|- ( k = j -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) ) |
156 |
155
|
adantl |
|- ( ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) /\ k = j ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) ) |
157 |
2
|
ffvelrnda |
|- ( ( ph /\ j e. X ) -> ( A ` j ) e. RR ) |
158 |
3
|
ffvelrnda |
|- ( ( ph /\ j e. X ) -> ( B ` j ) e. RR ) |
159 |
|
volico2 |
|- ( ( ( A ` j ) e. RR /\ ( B ` j ) e. RR ) -> ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) = if ( ( A ` j ) <_ ( B ` j ) , ( ( B ` j ) - ( A ` j ) ) , 0 ) ) |
160 |
157 158 159
|
syl2anc |
|- ( ( ph /\ j e. X ) -> ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) = if ( ( A ` j ) <_ ( B ` j ) , ( ( B ` j ) - ( A ` j ) ) , 0 ) ) |
161 |
160
|
3adant3 |
|- ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) -> ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) = if ( ( A ` j ) <_ ( B ` j ) , ( ( B ` j ) - ( A ` j ) ) , 0 ) ) |
162 |
|
simp3 |
|- ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) -> ( B ` j ) < ( A ` j ) ) |
163 |
158 157
|
ltnled |
|- ( ( ph /\ j e. X ) -> ( ( B ` j ) < ( A ` j ) <-> -. ( A ` j ) <_ ( B ` j ) ) ) |
164 |
163
|
3adant3 |
|- ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) -> ( ( B ` j ) < ( A ` j ) <-> -. ( A ` j ) <_ ( B ` j ) ) ) |
165 |
162 164
|
mpbid |
|- ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) -> -. ( A ` j ) <_ ( B ` j ) ) |
166 |
165
|
iffalsed |
|- ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) -> if ( ( A ` j ) <_ ( B ` j ) , ( ( B ` j ) - ( A ` j ) ) , 0 ) = 0 ) |
167 |
161 166
|
eqtrd |
|- ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) -> ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) = 0 ) |
168 |
167
|
adantr |
|- ( ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) /\ k = j ) -> ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) = 0 ) |
169 |
156 168
|
eqtrd |
|- ( ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) /\ k = j ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = 0 ) |
170 |
147 148 152 153 169
|
fprodeq0g |
|- ( ( ph /\ j e. X /\ ( B ` j ) < ( A ` j ) ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = 0 ) |
171 |
146 170
|
chvarvv |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = 0 ) |
172 |
141 171
|
eqtrd |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> ( A ( L ` X ) B ) = 0 ) |
173 |
121 137 172
|
3eqtr4d |
|- ( ( ph /\ k e. X /\ ( B ` k ) < ( A ` k ) ) -> ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) ) |
174 |
173
|
3exp |
|- ( ph -> ( k e. X -> ( ( B ` k ) < ( A ` k ) -> ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) ) ) ) |
175 |
174
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> ( k e. X -> ( ( B ` k ) < ( A ` k ) -> ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) ) ) ) |
176 |
34 118 175
|
rexlimd |
|- ( ( ph /\ X =/= (/) ) -> ( E. k e. X ( B ` k ) < ( A ` k ) -> ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) ) ) |
177 |
176
|
imp |
|- ( ( ( ph /\ X =/= (/) ) /\ E. k e. X ( B ` k ) < ( A ` k ) ) -> ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) ) |
178 |
98 177
|
syldan |
|- ( ( ( ph /\ X =/= (/) ) /\ -. A. k e. X ( A ` k ) <_ ( B ` k ) ) -> ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) ) |
179 |
84 178
|
pm2.61dan |
|- ( ( ph /\ X =/= (/) ) -> ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) ) |
180 |
33 179
|
syldan |
|- ( ( ph /\ -. X = (/) ) -> ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) ) |
181 |
31 180
|
pm2.61dan |
|- ( ph -> ( ( voln ` X ) ` I ) = ( A ( L ` X ) B ) ) |