Step |
Hyp |
Ref |
Expression |
1 |
|
vonsn.1 |
|- ( ph -> X e. Fin ) |
2 |
|
vonsn.2 |
|- ( ph -> A e. ( RR ^m X ) ) |
3 |
|
fveq2 |
|- ( X = (/) -> ( voln ` X ) = ( voln ` (/) ) ) |
4 |
3
|
fveq1d |
|- ( X = (/) -> ( ( voln ` X ) ` { A } ) = ( ( voln ` (/) ) ` { A } ) ) |
5 |
4
|
adantl |
|- ( ( ph /\ X = (/) ) -> ( ( voln ` X ) ` { A } ) = ( ( voln ` (/) ) ` { A } ) ) |
6 |
|
0fin |
|- (/) e. Fin |
7 |
6
|
a1i |
|- ( ( ph /\ X = (/) ) -> (/) e. Fin ) |
8 |
2
|
adantr |
|- ( ( ph /\ X = (/) ) -> A e. ( RR ^m X ) ) |
9 |
|
oveq2 |
|- ( X = (/) -> ( RR ^m X ) = ( RR ^m (/) ) ) |
10 |
9
|
adantl |
|- ( ( ph /\ X = (/) ) -> ( RR ^m X ) = ( RR ^m (/) ) ) |
11 |
8 10
|
eleqtrd |
|- ( ( ph /\ X = (/) ) -> A e. ( RR ^m (/) ) ) |
12 |
7 11
|
snvonmbl |
|- ( ( ph /\ X = (/) ) -> { A } e. dom ( voln ` (/) ) ) |
13 |
12
|
von0val |
|- ( ( ph /\ X = (/) ) -> ( ( voln ` (/) ) ` { A } ) = 0 ) |
14 |
5 13
|
eqtrd |
|- ( ( ph /\ X = (/) ) -> ( ( voln ` X ) ` { A } ) = 0 ) |
15 |
|
neqne |
|- ( -. X = (/) -> X =/= (/) ) |
16 |
15
|
adantl |
|- ( ( ph /\ -. X = (/) ) -> X =/= (/) ) |
17 |
2
|
rrxsnicc |
|- ( ph -> X_ k e. X ( ( A ` k ) [,] ( A ` k ) ) = { A } ) |
18 |
17
|
eqcomd |
|- ( ph -> { A } = X_ k e. X ( ( A ` k ) [,] ( A ` k ) ) ) |
19 |
18
|
fveq2d |
|- ( ph -> ( ( voln ` X ) ` { A } ) = ( ( voln ` X ) ` X_ k e. X ( ( A ` k ) [,] ( A ` k ) ) ) ) |
20 |
19
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> ( ( voln ` X ) ` { A } ) = ( ( voln ` X ) ` X_ k e. X ( ( A ` k ) [,] ( A ` k ) ) ) ) |
21 |
1
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> X e. Fin ) |
22 |
|
simpr |
|- ( ( ph /\ X =/= (/) ) -> X =/= (/) ) |
23 |
|
elmapi |
|- ( A e. ( RR ^m X ) -> A : X --> RR ) |
24 |
2 23
|
syl |
|- ( ph -> A : X --> RR ) |
25 |
24
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> A : X --> RR ) |
26 |
|
eqid |
|- X_ k e. X ( ( A ` k ) [,] ( A ` k ) ) = X_ k e. X ( ( A ` k ) [,] ( A ` k ) ) |
27 |
21 22 25 25 26
|
vonn0icc |
|- ( ( ph /\ X =/= (/) ) -> ( ( voln ` X ) ` X_ k e. X ( ( A ` k ) [,] ( A ` k ) ) ) = prod_ k e. X ( vol ` ( ( A ` k ) [,] ( A ` k ) ) ) ) |
28 |
24
|
ffvelrnda |
|- ( ( ph /\ k e. X ) -> ( A ` k ) e. RR ) |
29 |
28
|
rexrd |
|- ( ( ph /\ k e. X ) -> ( A ` k ) e. RR* ) |
30 |
|
iccid |
|- ( ( A ` k ) e. RR* -> ( ( A ` k ) [,] ( A ` k ) ) = { ( A ` k ) } ) |
31 |
29 30
|
syl |
|- ( ( ph /\ k e. X ) -> ( ( A ` k ) [,] ( A ` k ) ) = { ( A ` k ) } ) |
32 |
31
|
fveq2d |
|- ( ( ph /\ k e. X ) -> ( vol ` ( ( A ` k ) [,] ( A ` k ) ) ) = ( vol ` { ( A ` k ) } ) ) |
33 |
|
volsn |
|- ( ( A ` k ) e. RR -> ( vol ` { ( A ` k ) } ) = 0 ) |
34 |
28 33
|
syl |
|- ( ( ph /\ k e. X ) -> ( vol ` { ( A ` k ) } ) = 0 ) |
35 |
32 34
|
eqtrd |
|- ( ( ph /\ k e. X ) -> ( vol ` ( ( A ` k ) [,] ( A ` k ) ) ) = 0 ) |
36 |
35
|
prodeq2dv |
|- ( ph -> prod_ k e. X ( vol ` ( ( A ` k ) [,] ( A ` k ) ) ) = prod_ k e. X 0 ) |
37 |
36
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,] ( A ` k ) ) ) = prod_ k e. X 0 ) |
38 |
|
0cnd |
|- ( ph -> 0 e. CC ) |
39 |
|
fprodconst |
|- ( ( X e. Fin /\ 0 e. CC ) -> prod_ k e. X 0 = ( 0 ^ ( # ` X ) ) ) |
40 |
1 38 39
|
syl2anc |
|- ( ph -> prod_ k e. X 0 = ( 0 ^ ( # ` X ) ) ) |
41 |
40
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> prod_ k e. X 0 = ( 0 ^ ( # ` X ) ) ) |
42 |
|
hashnncl |
|- ( X e. Fin -> ( ( # ` X ) e. NN <-> X =/= (/) ) ) |
43 |
1 42
|
syl |
|- ( ph -> ( ( # ` X ) e. NN <-> X =/= (/) ) ) |
44 |
43
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> ( ( # ` X ) e. NN <-> X =/= (/) ) ) |
45 |
22 44
|
mpbird |
|- ( ( ph /\ X =/= (/) ) -> ( # ` X ) e. NN ) |
46 |
|
0exp |
|- ( ( # ` X ) e. NN -> ( 0 ^ ( # ` X ) ) = 0 ) |
47 |
45 46
|
syl |
|- ( ( ph /\ X =/= (/) ) -> ( 0 ^ ( # ` X ) ) = 0 ) |
48 |
37 41 47
|
3eqtrd |
|- ( ( ph /\ X =/= (/) ) -> prod_ k e. X ( vol ` ( ( A ` k ) [,] ( A ` k ) ) ) = 0 ) |
49 |
20 27 48
|
3eqtrd |
|- ( ( ph /\ X =/= (/) ) -> ( ( voln ` X ) ` { A } ) = 0 ) |
50 |
16 49
|
syldan |
|- ( ( ph /\ -. X = (/) ) -> ( ( voln ` X ) ` { A } ) = 0 ) |
51 |
14 50
|
pm2.61dan |
|- ( ph -> ( ( voln ` X ) ` { A } ) = 0 ) |