Step |
Hyp |
Ref |
Expression |
1 |
|
lkr0f.d |
|- D = ( Scalar ` W ) |
2 |
|
lkr0f.o |
|- .0. = ( 0g ` D ) |
3 |
|
lkr0f.v |
|- V = ( Base ` W ) |
4 |
|
lkr0f.f |
|- F = ( LFnl ` W ) |
5 |
|
lkr0f.k |
|- K = ( LKer ` W ) |
6 |
|
eqid |
|- ( Base ` D ) = ( Base ` D ) |
7 |
1 6 3 4
|
lflf |
|- ( ( W e. LMod /\ G e. F ) -> G : V --> ( Base ` D ) ) |
8 |
7
|
ffnd |
|- ( ( W e. LMod /\ G e. F ) -> G Fn V ) |
9 |
8
|
adantr |
|- ( ( ( W e. LMod /\ G e. F ) /\ ( K ` G ) = V ) -> G Fn V ) |
10 |
1 2 4 5
|
lkrval |
|- ( ( W e. LMod /\ G e. F ) -> ( K ` G ) = ( `' G " { .0. } ) ) |
11 |
10
|
eqeq1d |
|- ( ( W e. LMod /\ G e. F ) -> ( ( K ` G ) = V <-> ( `' G " { .0. } ) = V ) ) |
12 |
11
|
biimpa |
|- ( ( ( W e. LMod /\ G e. F ) /\ ( K ` G ) = V ) -> ( `' G " { .0. } ) = V ) |
13 |
2
|
fvexi |
|- .0. e. _V |
14 |
13
|
fconst2 |
|- ( G : V --> { .0. } <-> G = ( V X. { .0. } ) ) |
15 |
|
fconst4 |
|- ( G : V --> { .0. } <-> ( G Fn V /\ ( `' G " { .0. } ) = V ) ) |
16 |
14 15
|
bitr3i |
|- ( G = ( V X. { .0. } ) <-> ( G Fn V /\ ( `' G " { .0. } ) = V ) ) |
17 |
9 12 16
|
sylanbrc |
|- ( ( ( W e. LMod /\ G e. F ) /\ ( K ` G ) = V ) -> G = ( V X. { .0. } ) ) |
18 |
17
|
ex |
|- ( ( W e. LMod /\ G e. F ) -> ( ( K ` G ) = V -> G = ( V X. { .0. } ) ) ) |
19 |
16
|
biimpi |
|- ( G = ( V X. { .0. } ) -> ( G Fn V /\ ( `' G " { .0. } ) = V ) ) |
20 |
19
|
adantl |
|- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( G Fn V /\ ( `' G " { .0. } ) = V ) ) |
21 |
|
simpr |
|- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> G = ( V X. { .0. } ) ) |
22 |
|
eqid |
|- ( LFnl ` W ) = ( LFnl ` W ) |
23 |
1 2 3 22
|
lfl0f |
|- ( W e. LMod -> ( V X. { .0. } ) e. ( LFnl ` W ) ) |
24 |
23
|
adantr |
|- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( V X. { .0. } ) e. ( LFnl ` W ) ) |
25 |
21 24
|
eqeltrd |
|- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> G e. ( LFnl ` W ) ) |
26 |
1 2 22 5
|
lkrval |
|- ( ( W e. LMod /\ G e. ( LFnl ` W ) ) -> ( K ` G ) = ( `' G " { .0. } ) ) |
27 |
25 26
|
syldan |
|- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( K ` G ) = ( `' G " { .0. } ) ) |
28 |
27
|
eqeq1d |
|- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( ( K ` G ) = V <-> ( `' G " { .0. } ) = V ) ) |
29 |
|
ffn |
|- ( G : V --> { .0. } -> G Fn V ) |
30 |
14 29
|
sylbir |
|- ( G = ( V X. { .0. } ) -> G Fn V ) |
31 |
30
|
adantl |
|- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> G Fn V ) |
32 |
31
|
biantrurd |
|- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( ( `' G " { .0. } ) = V <-> ( G Fn V /\ ( `' G " { .0. } ) = V ) ) ) |
33 |
28 32
|
bitrd |
|- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( ( K ` G ) = V <-> ( G Fn V /\ ( `' G " { .0. } ) = V ) ) ) |
34 |
20 33
|
mpbird |
|- ( ( W e. LMod /\ G = ( V X. { .0. } ) ) -> ( K ` G ) = V ) |
35 |
34
|
ex |
|- ( W e. LMod -> ( G = ( V X. { .0. } ) -> ( K ` G ) = V ) ) |
36 |
35
|
adantr |
|- ( ( W e. LMod /\ G e. F ) -> ( G = ( V X. { .0. } ) -> ( K ` G ) = V ) ) |
37 |
18 36
|
impbid |
|- ( ( W e. LMod /\ G e. F ) -> ( ( K ` G ) = V <-> G = ( V X. { .0. } ) ) ) |