Step |
Hyp |
Ref |
Expression |
1 |
|
lshpset2.v |
|- V = ( Base ` W ) |
2 |
|
lshpset2.d |
|- D = ( Scalar ` W ) |
3 |
|
lshpset2.z |
|- .0. = ( 0g ` D ) |
4 |
|
lshpset2.h |
|- H = ( LSHyp ` W ) |
5 |
|
lshpset2.f |
|- F = ( LFnl ` W ) |
6 |
|
lshpset2.k |
|- K = ( LKer ` W ) |
7 |
4 5 6
|
lshpkrex |
|- ( ( W e. LVec /\ s e. H ) -> E. g e. F ( K ` g ) = s ) |
8 |
|
eleq1 |
|- ( ( K ` g ) = s -> ( ( K ` g ) e. H <-> s e. H ) ) |
9 |
8
|
biimparc |
|- ( ( s e. H /\ ( K ` g ) = s ) -> ( K ` g ) e. H ) |
10 |
9
|
adantll |
|- ( ( ( W e. LVec /\ s e. H ) /\ ( K ` g ) = s ) -> ( K ` g ) e. H ) |
11 |
10
|
adantlr |
|- ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> ( K ` g ) e. H ) |
12 |
|
simplll |
|- ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> W e. LVec ) |
13 |
|
simplr |
|- ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> g e. F ) |
14 |
1 2 3 4 5 6 12 13
|
lkrshp3 |
|- ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> ( ( K ` g ) e. H <-> g =/= ( V X. { .0. } ) ) ) |
15 |
11 14
|
mpbid |
|- ( ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) /\ ( K ` g ) = s ) -> g =/= ( V X. { .0. } ) ) |
16 |
15
|
ex |
|- ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) -> ( ( K ` g ) = s -> g =/= ( V X. { .0. } ) ) ) |
17 |
|
eqimss2 |
|- ( ( K ` g ) = s -> s C_ ( K ` g ) ) |
18 |
|
eqimss |
|- ( ( K ` g ) = s -> ( K ` g ) C_ s ) |
19 |
17 18
|
eqssd |
|- ( ( K ` g ) = s -> s = ( K ` g ) ) |
20 |
19
|
a1i |
|- ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) -> ( ( K ` g ) = s -> s = ( K ` g ) ) ) |
21 |
16 20
|
jcad |
|- ( ( ( W e. LVec /\ s e. H ) /\ g e. F ) -> ( ( K ` g ) = s -> ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) ) |
22 |
21
|
reximdva |
|- ( ( W e. LVec /\ s e. H ) -> ( E. g e. F ( K ` g ) = s -> E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) ) |
23 |
7 22
|
mpd |
|- ( ( W e. LVec /\ s e. H ) -> E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) |
24 |
23
|
ex |
|- ( W e. LVec -> ( s e. H -> E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) ) |
25 |
1 2 3 4 5 6
|
lkrshp |
|- ( ( W e. LVec /\ g e. F /\ g =/= ( V X. { .0. } ) ) -> ( K ` g ) e. H ) |
26 |
25
|
3adant3r |
|- ( ( W e. LVec /\ g e. F /\ ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) -> ( K ` g ) e. H ) |
27 |
|
eqid |
|- ( LSpan ` W ) = ( LSpan ` W ) |
28 |
|
eqid |
|- ( LSubSp ` W ) = ( LSubSp ` W ) |
29 |
1 27 28 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 ) ) ) |
30 |
29
|
3ad2ant1 |
|- ( ( W e. LVec /\ g e. F /\ ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) -> ( ( K ` g ) e. H <-> ( ( K ` g ) e. ( LSubSp ` W ) /\ ( K ` g ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) ) |
31 |
26 30
|
mpbid |
|- ( ( W e. LVec /\ g e. F /\ ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) -> ( ( K ` g ) e. ( LSubSp ` W ) /\ ( K ` g ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) |
32 |
|
eleq1 |
|- ( s = ( K ` g ) -> ( s e. ( LSubSp ` W ) <-> ( K ` g ) e. ( LSubSp ` W ) ) ) |
33 |
|
neeq1 |
|- ( s = ( K ` g ) -> ( s =/= V <-> ( K ` g ) =/= V ) ) |
34 |
|
uneq1 |
|- ( s = ( K ` g ) -> ( s u. { v } ) = ( ( K ` g ) u. { v } ) ) |
35 |
34
|
fveqeq2d |
|- ( s = ( K ` g ) -> ( ( ( LSpan ` W ) ` ( s u. { v } ) ) = V <-> ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) |
36 |
35
|
rexbidv |
|- ( s = ( K ` g ) -> ( E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V <-> E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) |
37 |
32 33 36
|
3anbi123d |
|- ( s = ( K ` g ) -> ( ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) <-> ( ( K ` g ) e. ( LSubSp ` W ) /\ ( K ` g ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) ) |
38 |
37
|
adantl |
|- ( ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) -> ( ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) <-> ( ( K ` g ) e. ( LSubSp ` W ) /\ ( K ` g ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) ) |
39 |
38
|
3ad2ant3 |
|- ( ( W e. LVec /\ g e. F /\ ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) -> ( ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) <-> ( ( K ` g ) e. ( LSubSp ` W ) /\ ( K ` g ) =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( ( K ` g ) u. { v } ) ) = V ) ) ) |
40 |
31 39
|
mpbird |
|- ( ( W e. LVec /\ g e. F /\ ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) -> ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) ) |
41 |
40
|
rexlimdv3a |
|- ( W e. LVec -> ( E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) -> ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) ) ) |
42 |
1 27 28 4
|
islshp |
|- ( W e. LVec -> ( s e. H <-> ( s e. ( LSubSp ` W ) /\ s =/= V /\ E. v e. V ( ( LSpan ` W ) ` ( s u. { v } ) ) = V ) ) ) |
43 |
41 42
|
sylibrd |
|- ( W e. LVec -> ( E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) -> s e. H ) ) |
44 |
24 43
|
impbid |
|- ( W e. LVec -> ( s e. H <-> E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) ) ) |
45 |
44
|
abbi2dv |
|- ( W e. LVec -> H = { s | E. g e. F ( g =/= ( V X. { .0. } ) /\ s = ( K ` g ) ) } ) |