Step |
Hyp |
Ref |
Expression |
1 |
|
lkrsc.v |
|- V = ( Base ` W ) |
2 |
|
lkrsc.d |
|- D = ( Scalar ` W ) |
3 |
|
lkrsc.k |
|- K = ( Base ` D ) |
4 |
|
lkrsc.t |
|- .x. = ( .r ` D ) |
5 |
|
lkrsc.f |
|- F = ( LFnl ` W ) |
6 |
|
lkrsc.l |
|- L = ( LKer ` W ) |
7 |
|
lkrsc.w |
|- ( ph -> W e. LVec ) |
8 |
|
lkrsc.g |
|- ( ph -> G e. F ) |
9 |
|
lkrsc.r |
|- ( ph -> R e. K ) |
10 |
|
lkrsc.o |
|- .0. = ( 0g ` D ) |
11 |
|
lkrsc.e |
|- ( ph -> R =/= .0. ) |
12 |
1
|
fvexi |
|- V e. _V |
13 |
12
|
a1i |
|- ( ph -> V e. _V ) |
14 |
2 3 1 5
|
lflf |
|- ( ( W e. LVec /\ G e. F ) -> G : V --> K ) |
15 |
7 8 14
|
syl2anc |
|- ( ph -> G : V --> K ) |
16 |
15
|
ffnd |
|- ( ph -> G Fn V ) |
17 |
|
eqidd |
|- ( ( ph /\ v e. V ) -> ( G ` v ) = ( G ` v ) ) |
18 |
13 9 16 17
|
ofc2 |
|- ( ( ph /\ v e. V ) -> ( ( G oF .x. ( V X. { R } ) ) ` v ) = ( ( G ` v ) .x. R ) ) |
19 |
18
|
eqeq1d |
|- ( ( ph /\ v e. V ) -> ( ( ( G oF .x. ( V X. { R } ) ) ` v ) = .0. <-> ( ( G ` v ) .x. R ) = .0. ) ) |
20 |
2
|
lvecdrng |
|- ( W e. LVec -> D e. DivRing ) |
21 |
7 20
|
syl |
|- ( ph -> D e. DivRing ) |
22 |
21
|
adantr |
|- ( ( ph /\ v e. V ) -> D e. DivRing ) |
23 |
7
|
adantr |
|- ( ( ph /\ v e. V ) -> W e. LVec ) |
24 |
8
|
adantr |
|- ( ( ph /\ v e. V ) -> G e. F ) |
25 |
|
simpr |
|- ( ( ph /\ v e. V ) -> v e. V ) |
26 |
2 3 1 5
|
lflcl |
|- ( ( W e. LVec /\ G e. F /\ v e. V ) -> ( G ` v ) e. K ) |
27 |
23 24 25 26
|
syl3anc |
|- ( ( ph /\ v e. V ) -> ( G ` v ) e. K ) |
28 |
9
|
adantr |
|- ( ( ph /\ v e. V ) -> R e. K ) |
29 |
11
|
adantr |
|- ( ( ph /\ v e. V ) -> R =/= .0. ) |
30 |
3 10 4 22 27 28 29
|
drngmuleq0 |
|- ( ( ph /\ v e. V ) -> ( ( ( G ` v ) .x. R ) = .0. <-> ( G ` v ) = .0. ) ) |
31 |
19 30
|
bitrd |
|- ( ( ph /\ v e. V ) -> ( ( ( G oF .x. ( V X. { R } ) ) ` v ) = .0. <-> ( G ` v ) = .0. ) ) |
32 |
31
|
pm5.32da |
|- ( ph -> ( ( v e. V /\ ( ( G oF .x. ( V X. { R } ) ) ` v ) = .0. ) <-> ( v e. V /\ ( G ` v ) = .0. ) ) ) |
33 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
34 |
7 33
|
syl |
|- ( ph -> W e. LMod ) |
35 |
1 2 3 4 5 34 8 9
|
lflvscl |
|- ( ph -> ( G oF .x. ( V X. { R } ) ) e. F ) |
36 |
1 2 10 5 6
|
ellkr |
|- ( ( W e. LVec /\ ( G oF .x. ( V X. { R } ) ) e. F ) -> ( v e. ( L ` ( G oF .x. ( V X. { R } ) ) ) <-> ( v e. V /\ ( ( G oF .x. ( V X. { R } ) ) ` v ) = .0. ) ) ) |
37 |
7 35 36
|
syl2anc |
|- ( ph -> ( v e. ( L ` ( G oF .x. ( V X. { R } ) ) ) <-> ( v e. V /\ ( ( G oF .x. ( V X. { R } ) ) ` v ) = .0. ) ) ) |
38 |
1 2 10 5 6
|
ellkr |
|- ( ( W e. LVec /\ G e. F ) -> ( v e. ( L ` G ) <-> ( v e. V /\ ( G ` v ) = .0. ) ) ) |
39 |
7 8 38
|
syl2anc |
|- ( ph -> ( v e. ( L ` G ) <-> ( v e. V /\ ( G ` v ) = .0. ) ) ) |
40 |
32 37 39
|
3bitr4d |
|- ( ph -> ( v e. ( L ` ( G oF .x. ( V X. { R } ) ) ) <-> v e. ( L ` G ) ) ) |
41 |
40
|
eqrdv |
|- ( ph -> ( L ` ( G oF .x. ( V X. { R } ) ) ) = ( L ` G ) ) |