Step |
Hyp |
Ref |
Expression |
1 |
|
hoidmvval0.p |
|- F/ j ph |
2 |
|
hoidmvval0.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 ) ) ) ) ) ) |
3 |
|
hoidmvval0.x |
|- ( ph -> X e. Fin ) |
4 |
|
hoidmvval0.a |
|- ( ph -> A : X --> RR ) |
5 |
|
hoidmvval0.b |
|- ( ph -> B : X --> RR ) |
6 |
|
hoidmvval0.j |
|- ( ph -> E. j e. X ( B ` j ) <_ ( A ` j ) ) |
7 |
|
id |
|- ( ph -> ph ) |
8 |
|
fveq2 |
|- ( k = j -> ( B ` k ) = ( B ` j ) ) |
9 |
|
fveq2 |
|- ( k = j -> ( A ` k ) = ( A ` j ) ) |
10 |
8 9
|
breq12d |
|- ( k = j -> ( ( B ` k ) <_ ( A ` k ) <-> ( B ` j ) <_ ( A ` j ) ) ) |
11 |
10
|
cbvrexvw |
|- ( E. k e. X ( B ` k ) <_ ( A ` k ) <-> E. j e. X ( B ` j ) <_ ( A ` j ) ) |
12 |
|
rexn0 |
|- ( E. k e. X ( B ` k ) <_ ( A ` k ) -> X =/= (/) ) |
13 |
11 12
|
sylbir |
|- ( E. j e. X ( B ` j ) <_ ( A ` j ) -> X =/= (/) ) |
14 |
6 13
|
syl |
|- ( ph -> X =/= (/) ) |
15 |
3
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> X e. Fin ) |
16 |
|
simpr |
|- ( ( ph /\ X =/= (/) ) -> X =/= (/) ) |
17 |
4
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> A : X --> RR ) |
18 |
5
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> B : X --> RR ) |
19 |
2 15 16 17 18
|
hoidmvn0val |
|- ( ( ph /\ X =/= (/) ) -> ( A ( L ` X ) B ) = prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) ) |
20 |
6
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> E. j e. X ( B ` j ) <_ ( A ` j ) ) |
21 |
|
nfv |
|- F/ j X =/= (/) |
22 |
1 21
|
nfan |
|- F/ j ( ph /\ X =/= (/) ) |
23 |
|
nfv |
|- F/ j prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = 0 |
24 |
|
nfv |
|- F/ k ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) |
25 |
|
nfcv |
|- F/_ k ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) |
26 |
3
|
3ad2ant1 |
|- ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> X e. Fin ) |
27 |
4
|
ffvelrnda |
|- ( ( ph /\ k e. X ) -> ( A ` k ) e. RR ) |
28 |
5
|
ffvelrnda |
|- ( ( ph /\ k e. X ) -> ( B ` k ) e. RR ) |
29 |
|
volicore |
|- ( ( ( A ` k ) e. RR /\ ( B ` k ) e. RR ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) e. RR ) |
30 |
27 28 29
|
syl2anc |
|- ( ( ph /\ k e. X ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) e. RR ) |
31 |
30
|
recnd |
|- ( ( ph /\ k e. X ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) e. CC ) |
32 |
31
|
3ad2antl1 |
|- ( ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) /\ k e. X ) -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) e. CC ) |
33 |
9 8
|
oveq12d |
|- ( k = j -> ( ( A ` k ) [,) ( B ` k ) ) = ( ( A ` j ) [,) ( B ` j ) ) ) |
34 |
33
|
fveq2d |
|- ( k = j -> ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) ) |
35 |
|
simp2 |
|- ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> j e. X ) |
36 |
4
|
ffvelrnda |
|- ( ( ph /\ j e. X ) -> ( A ` j ) e. RR ) |
37 |
36
|
3adant3 |
|- ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> ( A ` j ) e. RR ) |
38 |
5
|
ffvelrnda |
|- ( ( ph /\ j e. X ) -> ( B ` j ) e. RR ) |
39 |
38
|
3adant3 |
|- ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> ( B ` j ) e. RR ) |
40 |
|
volico |
|- ( ( ( A ` j ) e. RR /\ ( B ` j ) e. RR ) -> ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) = if ( ( A ` j ) < ( B ` j ) , ( ( B ` j ) - ( A ` j ) ) , 0 ) ) |
41 |
37 39 40
|
syl2anc |
|- ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) = if ( ( A ` j ) < ( B ` j ) , ( ( B ` j ) - ( A ` j ) ) , 0 ) ) |
42 |
|
simp3 |
|- ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> ( B ` j ) <_ ( A ` j ) ) |
43 |
39 37
|
lenltd |
|- ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> ( ( B ` j ) <_ ( A ` j ) <-> -. ( A ` j ) < ( B ` j ) ) ) |
44 |
42 43
|
mpbid |
|- ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> -. ( A ` j ) < ( B ` j ) ) |
45 |
44
|
iffalsed |
|- ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> if ( ( A ` j ) < ( B ` j ) , ( ( B ` j ) - ( A ` j ) ) , 0 ) = 0 ) |
46 |
41 45
|
eqtrd |
|- ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> ( vol ` ( ( A ` j ) [,) ( B ` j ) ) ) = 0 ) |
47 |
24 25 26 32 34 35 46
|
fprod0 |
|- ( ( ph /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = 0 ) |
48 |
47
|
3adant1r |
|- ( ( ( ph /\ X =/= (/) ) /\ j e. X /\ ( B ` j ) <_ ( A ` j ) ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = 0 ) |
49 |
48
|
3exp |
|- ( ( ph /\ X =/= (/) ) -> ( j e. X -> ( ( B ` j ) <_ ( A ` j ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = 0 ) ) ) |
50 |
22 23 49
|
rexlimd |
|- ( ( ph /\ X =/= (/) ) -> ( E. j e. X ( B ` j ) <_ ( A ` j ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = 0 ) ) |
51 |
20 50
|
mpd |
|- ( ( ph /\ X =/= (/) ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,) ( B ` k ) ) ) = 0 ) |
52 |
|
eqidd |
|- ( ( ph /\ X =/= (/) ) -> 0 = 0 ) |
53 |
19 51 52
|
3eqtrd |
|- ( ( ph /\ X =/= (/) ) -> ( A ( L ` X ) B ) = 0 ) |
54 |
7 14 53
|
syl2anc |
|- ( ph -> ( A ( L ` X ) B ) = 0 ) |