Step |
Hyp |
Ref |
Expression |
1 |
|
lfl1dim.v |
|- V = ( Base ` W ) |
2 |
|
lfl1dim.d |
|- D = ( Scalar ` W ) |
3 |
|
lfl1dim.f |
|- F = ( LFnl ` W ) |
4 |
|
lfl1dim.l |
|- L = ( LKer ` W ) |
5 |
|
lfl1dim.k |
|- K = ( Base ` D ) |
6 |
|
lfl1dim.t |
|- .x. = ( .r ` D ) |
7 |
|
lfl1dim.w |
|- ( ph -> W e. LVec ) |
8 |
|
lfl1dim.g |
|- ( ph -> G e. F ) |
9 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
10 |
7 9
|
syl |
|- ( ph -> W e. LMod ) |
11 |
|
eqid |
|- ( 0g ` D ) = ( 0g ` D ) |
12 |
2 5 11
|
lmod0cl |
|- ( W e. LMod -> ( 0g ` D ) e. K ) |
13 |
10 12
|
syl |
|- ( ph -> ( 0g ` D ) e. K ) |
14 |
13
|
ad2antrr |
|- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> ( 0g ` D ) e. K ) |
15 |
|
simpr |
|- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> g = ( V X. { ( 0g ` D ) } ) ) |
16 |
10
|
ad2antrr |
|- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> W e. LMod ) |
17 |
8
|
ad2antrr |
|- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> G e. F ) |
18 |
1 2 3 5 6 11 16 17
|
lfl0sc |
|- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> ( G oF .x. ( V X. { ( 0g ` D ) } ) ) = ( V X. { ( 0g ` D ) } ) ) |
19 |
15 18
|
eqtr4d |
|- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> g = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) |
20 |
|
sneq |
|- ( k = ( 0g ` D ) -> { k } = { ( 0g ` D ) } ) |
21 |
20
|
xpeq2d |
|- ( k = ( 0g ` D ) -> ( V X. { k } ) = ( V X. { ( 0g ` D ) } ) ) |
22 |
21
|
oveq2d |
|- ( k = ( 0g ` D ) -> ( G oF .x. ( V X. { k } ) ) = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) |
23 |
22
|
rspceeqv |
|- ( ( ( 0g ` D ) e. K /\ g = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) |
24 |
14 19 23
|
syl2anc |
|- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) |
25 |
24
|
a1d |
|- ( ( ( ph /\ g e. F ) /\ g = ( V X. { ( 0g ` D ) } ) ) -> ( ( L ` G ) C_ ( L ` g ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) ) |
26 |
13
|
ad3antrrr |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> ( 0g ` D ) e. K ) |
27 |
10
|
ad3antrrr |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> W e. LMod ) |
28 |
|
simpllr |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> g e. F ) |
29 |
1 3 4 27 28
|
lkrssv |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> ( L ` g ) C_ V ) |
30 |
10
|
adantr |
|- ( ( ph /\ g e. F ) -> W e. LMod ) |
31 |
8
|
adantr |
|- ( ( ph /\ g e. F ) -> G e. F ) |
32 |
2 11 1 3 4
|
lkr0f |
|- ( ( W e. LMod /\ G e. F ) -> ( ( L ` G ) = V <-> G = ( V X. { ( 0g ` D ) } ) ) ) |
33 |
30 31 32
|
syl2anc |
|- ( ( ph /\ g e. F ) -> ( ( L ` G ) = V <-> G = ( V X. { ( 0g ` D ) } ) ) ) |
34 |
33
|
biimpar |
|- ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) -> ( L ` G ) = V ) |
35 |
34
|
sseq1d |
|- ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) -> ( ( L ` G ) C_ ( L ` g ) <-> V C_ ( L ` g ) ) ) |
36 |
35
|
biimpa |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> V C_ ( L ` g ) ) |
37 |
29 36
|
eqssd |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> ( L ` g ) = V ) |
38 |
2 11 1 3 4
|
lkr0f |
|- ( ( W e. LMod /\ g e. F ) -> ( ( L ` g ) = V <-> g = ( V X. { ( 0g ` D ) } ) ) ) |
39 |
27 28 38
|
syl2anc |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> ( ( L ` g ) = V <-> g = ( V X. { ( 0g ` D ) } ) ) ) |
40 |
37 39
|
mpbid |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> g = ( V X. { ( 0g ` D ) } ) ) |
41 |
8
|
ad3antrrr |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> G e. F ) |
42 |
1 2 3 5 6 11 27 41
|
lfl0sc |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> ( G oF .x. ( V X. { ( 0g ` D ) } ) ) = ( V X. { ( 0g ` D ) } ) ) |
43 |
40 42
|
eqtr4d |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> g = ( G oF .x. ( V X. { ( 0g ` D ) } ) ) ) |
44 |
26 43 23
|
syl2anc |
|- ( ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) /\ ( L ` G ) C_ ( L ` g ) ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) |
45 |
44
|
ex |
|- ( ( ( ph /\ g e. F ) /\ G = ( V X. { ( 0g ` D ) } ) ) -> ( ( L ` G ) C_ ( L ` g ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) ) |
46 |
|
eqid |
|- ( LSHyp ` W ) = ( LSHyp ` W ) |
47 |
7
|
ad2antrr |
|- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> W e. LVec ) |
48 |
8
|
ad2antrr |
|- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> G e. F ) |
49 |
|
simprr |
|- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> G =/= ( V X. { ( 0g ` D ) } ) ) |
50 |
1 2 11 46 3 4
|
lkrshp |
|- ( ( W e. LVec /\ G e. F /\ G =/= ( V X. { ( 0g ` D ) } ) ) -> ( L ` G ) e. ( LSHyp ` W ) ) |
51 |
47 48 49 50
|
syl3anc |
|- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> ( L ` G ) e. ( LSHyp ` W ) ) |
52 |
|
simplr |
|- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> g e. F ) |
53 |
|
simprl |
|- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> g =/= ( V X. { ( 0g ` D ) } ) ) |
54 |
1 2 11 46 3 4
|
lkrshp |
|- ( ( W e. LVec /\ g e. F /\ g =/= ( V X. { ( 0g ` D ) } ) ) -> ( L ` g ) e. ( LSHyp ` W ) ) |
55 |
47 52 53 54
|
syl3anc |
|- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> ( L ` g ) e. ( LSHyp ` W ) ) |
56 |
46 47 51 55
|
lshpcmp |
|- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> ( ( L ` G ) C_ ( L ` g ) <-> ( L ` G ) = ( L ` g ) ) ) |
57 |
7
|
ad3antrrr |
|- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> W e. LVec ) |
58 |
8
|
ad3antrrr |
|- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> G e. F ) |
59 |
|
simpllr |
|- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> g e. F ) |
60 |
|
simpr |
|- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> ( L ` G ) = ( L ` g ) ) |
61 |
2 5 6 1 3 4
|
eqlkr2 |
|- ( ( W e. LVec /\ ( G e. F /\ g e. F ) /\ ( L ` G ) = ( L ` g ) ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) |
62 |
57 58 59 60 61
|
syl121anc |
|- ( ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) /\ ( L ` G ) = ( L ` g ) ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) |
63 |
62
|
ex |
|- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> ( ( L ` G ) = ( L ` g ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) ) |
64 |
56 63
|
sylbid |
|- ( ( ( ph /\ g e. F ) /\ ( g =/= ( V X. { ( 0g ` D ) } ) /\ G =/= ( V X. { ( 0g ` D ) } ) ) ) -> ( ( L ` G ) C_ ( L ` g ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) ) |
65 |
25 45 64
|
pm2.61da2ne |
|- ( ( ph /\ g e. F ) -> ( ( L ` G ) C_ ( L ` g ) -> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) ) |
66 |
7
|
ad2antrr |
|- ( ( ( ph /\ g e. F ) /\ k e. K ) -> W e. LVec ) |
67 |
8
|
ad2antrr |
|- ( ( ( ph /\ g e. F ) /\ k e. K ) -> G e. F ) |
68 |
|
simpr |
|- ( ( ( ph /\ g e. F ) /\ k e. K ) -> k e. K ) |
69 |
1 2 5 6 3 4 66 67 68
|
lkrscss |
|- ( ( ( ph /\ g e. F ) /\ k e. K ) -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { k } ) ) ) ) |
70 |
69
|
ex |
|- ( ( ph /\ g e. F ) -> ( k e. K -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { k } ) ) ) ) ) |
71 |
|
fveq2 |
|- ( g = ( G oF .x. ( V X. { k } ) ) -> ( L ` g ) = ( L ` ( G oF .x. ( V X. { k } ) ) ) ) |
72 |
71
|
sseq2d |
|- ( g = ( G oF .x. ( V X. { k } ) ) -> ( ( L ` G ) C_ ( L ` g ) <-> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { k } ) ) ) ) ) |
73 |
72
|
biimprcd |
|- ( ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { k } ) ) ) -> ( g = ( G oF .x. ( V X. { k } ) ) -> ( L ` G ) C_ ( L ` g ) ) ) |
74 |
70 73
|
syl6 |
|- ( ( ph /\ g e. F ) -> ( k e. K -> ( g = ( G oF .x. ( V X. { k } ) ) -> ( L ` G ) C_ ( L ` g ) ) ) ) |
75 |
74
|
rexlimdv |
|- ( ( ph /\ g e. F ) -> ( E. k e. K g = ( G oF .x. ( V X. { k } ) ) -> ( L ` G ) C_ ( L ` g ) ) ) |
76 |
65 75
|
impbid |
|- ( ( ph /\ g e. F ) -> ( ( L ` G ) C_ ( L ` g ) <-> E. k e. K g = ( G oF .x. ( V X. { k } ) ) ) ) |
77 |
76
|
rabbidva |
|- ( ph -> { g e. F | ( L ` G ) C_ ( L ` g ) } = { g e. F | E. k e. K g = ( G oF .x. ( V X. { k } ) ) } ) |