Step |
Hyp |
Ref |
Expression |
1 |
|
lkrshp.v |
|- V = ( Base ` W ) |
2 |
|
lkrshp.d |
|- D = ( Scalar ` W ) |
3 |
|
lkrshp.z |
|- .0. = ( 0g ` D ) |
4 |
|
lkrshp.h |
|- H = ( LSHyp ` W ) |
5 |
|
lkrshp.f |
|- F = ( LFnl ` W ) |
6 |
|
lkrshp.k |
|- K = ( LKer ` W ) |
7 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
8 |
7
|
3ad2ant1 |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> W e. LMod ) |
9 |
|
simp2 |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> G e. F ) |
10 |
|
eqid |
|- ( LSubSp ` W ) = ( LSubSp ` W ) |
11 |
5 6 10
|
lkrlss |
|- ( ( W e. LMod /\ G e. F ) -> ( K ` G ) e. ( LSubSp ` W ) ) |
12 |
8 9 11
|
syl2anc |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> ( K ` G ) e. ( LSubSp ` W ) ) |
13 |
|
simp3 |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> G =/= ( V X. { .0. } ) ) |
14 |
2 3 1 5 6
|
lkr0f |
|- ( ( W e. LMod /\ G e. F ) -> ( ( K ` G ) = V <-> G = ( V X. { .0. } ) ) ) |
15 |
8 9 14
|
syl2anc |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> ( ( K ` G ) = V <-> G = ( V X. { .0. } ) ) ) |
16 |
15
|
necon3bid |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> ( ( K ` G ) =/= V <-> G =/= ( V X. { .0. } ) ) ) |
17 |
13 16
|
mpbird |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> ( K ` G ) =/= V ) |
18 |
|
eqid |
|- ( 1r ` D ) = ( 1r ` D ) |
19 |
2 3 18 1 5
|
lfl1 |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> E. v e. V ( G ` v ) = ( 1r ` D ) ) |
20 |
|
simp11 |
|- ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) -> W e. LVec ) |
21 |
|
simp2 |
|- ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) -> v e. V ) |
22 |
|
simp12 |
|- ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) -> G e. F ) |
23 |
|
simp3 |
|- ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) -> ( G ` v ) = ( 1r ` D ) ) |
24 |
2
|
lvecdrng |
|- ( W e. LVec -> D e. DivRing ) |
25 |
3 18
|
drngunz |
|- ( D e. DivRing -> ( 1r ` D ) =/= .0. ) |
26 |
20 24 25
|
3syl |
|- ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) -> ( 1r ` D ) =/= .0. ) |
27 |
23 26
|
eqnetrd |
|- ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) -> ( G ` v ) =/= .0. ) |
28 |
|
simpl11 |
|- ( ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) /\ v e. ( K ` G ) ) -> W e. LVec ) |
29 |
|
simpl12 |
|- ( ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) /\ v e. ( K ` G ) ) -> G e. F ) |
30 |
|
simpr |
|- ( ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) /\ v e. ( K ` G ) ) -> v e. ( K ` G ) ) |
31 |
2 3 5 6
|
lkrf0 |
|- ( ( W e. LVec /\ G e. F /\ v e. ( K ` G ) ) -> ( G ` v ) = .0. ) |
32 |
28 29 30 31
|
syl3anc |
|- ( ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) /\ v e. ( K ` G ) ) -> ( G ` v ) = .0. ) |
33 |
32
|
ex |
|- ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) -> ( v e. ( K ` G ) -> ( G ` v ) = .0. ) ) |
34 |
33
|
necon3ad |
|- ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) -> ( ( G ` v ) =/= .0. -> -. v e. ( K ` G ) ) ) |
35 |
27 34
|
mpd |
|- ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) -> -. v e. ( K ` G ) ) |
36 |
|
eqid |
|- ( LSpan ` W ) = ( LSpan ` W ) |
37 |
1 36 5 6
|
lkrlsp3 |
|- ( ( W e. LVec /\ ( v e. V /\ G e. F ) /\ -. v e. ( K ` G ) ) -> ( ( LSpan ` W ) ` ( ( K ` G ) u. { v } ) ) = V ) |
38 |
20 21 22 35 37
|
syl121anc |
|- ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V /\ ( G ` v ) = ( 1r ` D ) ) -> ( ( LSpan ` W ) ` ( ( K ` G ) u. { v } ) ) = V ) |
39 |
38
|
3expia |
|- ( ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) /\ v e. V ) -> ( ( G ` v ) = ( 1r ` D ) -> ( ( LSpan ` W ) ` ( ( K ` G ) u. { v } ) ) = V ) ) |
40 |
39
|
reximdva |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> ( E. v e. V ( G ` v ) = ( 1r ` D ) -> E. v e. V ( ( LSpan ` W ) ` ( ( K ` G ) u. { v } ) ) = V ) ) |
41 |
19 40
|
mpd |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> E. v e. V ( ( LSpan ` W ) ` ( ( K ` G ) u. { v } ) ) = V ) |
42 |
1 36 10 4
|
islshp |
|- ( W e. LVec -> ( ( K ` G ) e. H <-> ( ( K ` G ) e. ( LSubSp ` W ) /\ ( K ` G ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` G ) u. { v } ) ) = V ) ) ) |
43 |
42
|
3ad2ant1 |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> ( ( K ` G ) e. H <-> ( ( K ` G ) e. ( LSubSp ` W ) /\ ( K ` G ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` G ) u. { v } ) ) = V ) ) ) |
44 |
12 17 41 43
|
mpbir3and |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { .0. } ) ) -> ( K ` G ) e. H ) |