Step |
Hyp |
Ref |
Expression |
1 |
|
lshpdisj.v |
|- V = ( Base ` W ) |
2 |
|
lshpdisj.o |
|- .0. = ( 0g ` W ) |
3 |
|
lshpdisj.n |
|- N = ( LSpan ` W ) |
4 |
|
lshpdisj.p |
|- .(+) = ( LSSum ` W ) |
5 |
|
lshpdisj.h |
|- H = ( LSHyp ` W ) |
6 |
|
lshpdisj.w |
|- ( ph -> W e. LVec ) |
7 |
|
lshpdisj.u |
|- ( ph -> U e. H ) |
8 |
|
lshpdisj.x |
|- ( ph -> X e. V ) |
9 |
|
lshpdisj.e |
|- ( ph -> ( U .(+) ( N ` { X } ) ) = V ) |
10 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
11 |
6 10
|
syl |
|- ( ph -> W e. LMod ) |
12 |
11
|
adantr |
|- ( ( ph /\ v e. U ) -> W e. LMod ) |
13 |
8
|
adantr |
|- ( ( ph /\ v e. U ) -> X e. V ) |
14 |
|
eqid |
|- ( Scalar ` W ) = ( Scalar ` W ) |
15 |
|
eqid |
|- ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) ) |
16 |
|
eqid |
|- ( .s ` W ) = ( .s ` W ) |
17 |
14 15 1 16 3
|
lspsnel |
|- ( ( W e. LMod /\ X e. V ) -> ( v e. ( N ` { X } ) <-> E. k e. ( Base ` ( Scalar ` W ) ) v = ( k ( .s ` W ) X ) ) ) |
18 |
12 13 17
|
syl2anc |
|- ( ( ph /\ v e. U ) -> ( v e. ( N ` { X } ) <-> E. k e. ( Base ` ( Scalar ` W ) ) v = ( k ( .s ` W ) X ) ) ) |
19 |
1 3 4 5 11 7 8 9
|
lshpnel |
|- ( ph -> -. X e. U ) |
20 |
19
|
ad2antrr |
|- ( ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) /\ ( k ( .s ` W ) X ) =/= .0. ) -> -. X e. U ) |
21 |
|
eqid |
|- ( LSubSp ` W ) = ( LSubSp ` W ) |
22 |
6
|
ad2antrr |
|- ( ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) /\ ( k ( .s ` W ) X ) =/= .0. ) -> W e. LVec ) |
23 |
21 5 11 7
|
lshplss |
|- ( ph -> U e. ( LSubSp ` W ) ) |
24 |
23
|
ad2antrr |
|- ( ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) /\ ( k ( .s ` W ) X ) =/= .0. ) -> U e. ( LSubSp ` W ) ) |
25 |
8
|
ad2antrr |
|- ( ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) /\ ( k ( .s ` W ) X ) =/= .0. ) -> X e. V ) |
26 |
11
|
adantr |
|- ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) -> W e. LMod ) |
27 |
|
simpr |
|- ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) -> k e. ( Base ` ( Scalar ` W ) ) ) |
28 |
8
|
adantr |
|- ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) -> X e. V ) |
29 |
1 16 14 15 3 26 27 28
|
lspsneli |
|- ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) -> ( k ( .s ` W ) X ) e. ( N ` { X } ) ) |
30 |
29
|
adantr |
|- ( ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) /\ ( k ( .s ` W ) X ) =/= .0. ) -> ( k ( .s ` W ) X ) e. ( N ` { X } ) ) |
31 |
|
simpr |
|- ( ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) /\ ( k ( .s ` W ) X ) =/= .0. ) -> ( k ( .s ` W ) X ) =/= .0. ) |
32 |
1 2 21 3 22 24 25 30 31
|
lspsnel4 |
|- ( ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) /\ ( k ( .s ` W ) X ) =/= .0. ) -> ( X e. U <-> ( k ( .s ` W ) X ) e. U ) ) |
33 |
20 32
|
mtbid |
|- ( ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) /\ ( k ( .s ` W ) X ) =/= .0. ) -> -. ( k ( .s ` W ) X ) e. U ) |
34 |
33
|
ex |
|- ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) -> ( ( k ( .s ` W ) X ) =/= .0. -> -. ( k ( .s ` W ) X ) e. U ) ) |
35 |
34
|
necon4ad |
|- ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) -> ( ( k ( .s ` W ) X ) e. U -> ( k ( .s ` W ) X ) = .0. ) ) |
36 |
|
eleq1 |
|- ( v = ( k ( .s ` W ) X ) -> ( v e. U <-> ( k ( .s ` W ) X ) e. U ) ) |
37 |
|
eqeq1 |
|- ( v = ( k ( .s ` W ) X ) -> ( v = .0. <-> ( k ( .s ` W ) X ) = .0. ) ) |
38 |
36 37
|
imbi12d |
|- ( v = ( k ( .s ` W ) X ) -> ( ( v e. U -> v = .0. ) <-> ( ( k ( .s ` W ) X ) e. U -> ( k ( .s ` W ) X ) = .0. ) ) ) |
39 |
35 38
|
syl5ibrcom |
|- ( ( ph /\ k e. ( Base ` ( Scalar ` W ) ) ) -> ( v = ( k ( .s ` W ) X ) -> ( v e. U -> v = .0. ) ) ) |
40 |
39
|
ex |
|- ( ph -> ( k e. ( Base ` ( Scalar ` W ) ) -> ( v = ( k ( .s ` W ) X ) -> ( v e. U -> v = .0. ) ) ) ) |
41 |
40
|
com23 |
|- ( ph -> ( v = ( k ( .s ` W ) X ) -> ( k e. ( Base ` ( Scalar ` W ) ) -> ( v e. U -> v = .0. ) ) ) ) |
42 |
41
|
com24 |
|- ( ph -> ( v e. U -> ( k e. ( Base ` ( Scalar ` W ) ) -> ( v = ( k ( .s ` W ) X ) -> v = .0. ) ) ) ) |
43 |
42
|
imp31 |
|- ( ( ( ph /\ v e. U ) /\ k e. ( Base ` ( Scalar ` W ) ) ) -> ( v = ( k ( .s ` W ) X ) -> v = .0. ) ) |
44 |
43
|
rexlimdva |
|- ( ( ph /\ v e. U ) -> ( E. k e. ( Base ` ( Scalar ` W ) ) v = ( k ( .s ` W ) X ) -> v = .0. ) ) |
45 |
18 44
|
sylbid |
|- ( ( ph /\ v e. U ) -> ( v e. ( N ` { X } ) -> v = .0. ) ) |
46 |
45
|
expimpd |
|- ( ph -> ( ( v e. U /\ v e. ( N ` { X } ) ) -> v = .0. ) ) |
47 |
|
elin |
|- ( v e. ( U i^i ( N ` { X } ) ) <-> ( v e. U /\ v e. ( N ` { X } ) ) ) |
48 |
|
velsn |
|- ( v e. { .0. } <-> v = .0. ) |
49 |
46 47 48
|
3imtr4g |
|- ( ph -> ( v e. ( U i^i ( N ` { X } ) ) -> v e. { .0. } ) ) |
50 |
49
|
ssrdv |
|- ( ph -> ( U i^i ( N ` { X } ) ) C_ { .0. } ) |
51 |
1 21 3
|
lspsncl |
|- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` W ) ) |
52 |
11 8 51
|
syl2anc |
|- ( ph -> ( N ` { X } ) e. ( LSubSp ` W ) ) |
53 |
21
|
lssincl |
|- ( ( W e. LMod /\ U e. ( LSubSp ` W ) /\ ( N ` { X } ) e. ( LSubSp ` W ) ) -> ( U i^i ( N ` { X } ) ) e. ( LSubSp ` W ) ) |
54 |
11 23 52 53
|
syl3anc |
|- ( ph -> ( U i^i ( N ` { X } ) ) e. ( LSubSp ` W ) ) |
55 |
2 21
|
lss0ss |
|- ( ( W e. LMod /\ ( U i^i ( N ` { X } ) ) e. ( LSubSp ` W ) ) -> { .0. } C_ ( U i^i ( N ` { X } ) ) ) |
56 |
11 54 55
|
syl2anc |
|- ( ph -> { .0. } C_ ( U i^i ( N ` { X } ) ) ) |
57 |
50 56
|
eqssd |
|- ( ph -> ( U i^i ( N ` { X } ) ) = { .0. } ) |