Step |
Hyp |
Ref |
Expression |
1 |
|
lspdisj2.v |
|- V = ( Base ` W ) |
2 |
|
lspdisj2.o |
|- .0. = ( 0g ` W ) |
3 |
|
lspdisj2.n |
|- N = ( LSpan ` W ) |
4 |
|
lspdisj2.w |
|- ( ph -> W e. LVec ) |
5 |
|
lspdisj2.x |
|- ( ph -> X e. V ) |
6 |
|
lspdisj2.y |
|- ( ph -> Y e. V ) |
7 |
|
lspdisj2.q |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
8 |
|
sneq |
|- ( X = .0. -> { X } = { .0. } ) |
9 |
8
|
fveq2d |
|- ( X = .0. -> ( N ` { X } ) = ( N ` { .0. } ) ) |
10 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
11 |
4 10
|
syl |
|- ( ph -> W e. LMod ) |
12 |
2 3
|
lspsn0 |
|- ( W e. LMod -> ( N ` { .0. } ) = { .0. } ) |
13 |
11 12
|
syl |
|- ( ph -> ( N ` { .0. } ) = { .0. } ) |
14 |
9 13
|
sylan9eqr |
|- ( ( ph /\ X = .0. ) -> ( N ` { X } ) = { .0. } ) |
15 |
14
|
ineq1d |
|- ( ( ph /\ X = .0. ) -> ( ( N ` { X } ) i^i ( N ` { Y } ) ) = ( { .0. } i^i ( N ` { Y } ) ) ) |
16 |
|
eqid |
|- ( LSubSp ` W ) = ( LSubSp ` W ) |
17 |
1 16 3
|
lspsncl |
|- ( ( W e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` W ) ) |
18 |
11 6 17
|
syl2anc |
|- ( ph -> ( N ` { Y } ) e. ( LSubSp ` W ) ) |
19 |
2 16
|
lss0ss |
|- ( ( W e. LMod /\ ( N ` { Y } ) e. ( LSubSp ` W ) ) -> { .0. } C_ ( N ` { Y } ) ) |
20 |
11 18 19
|
syl2anc |
|- ( ph -> { .0. } C_ ( N ` { Y } ) ) |
21 |
|
df-ss |
|- ( { .0. } C_ ( N ` { Y } ) <-> ( { .0. } i^i ( N ` { Y } ) ) = { .0. } ) |
22 |
20 21
|
sylib |
|- ( ph -> ( { .0. } i^i ( N ` { Y } ) ) = { .0. } ) |
23 |
22
|
adantr |
|- ( ( ph /\ X = .0. ) -> ( { .0. } i^i ( N ` { Y } ) ) = { .0. } ) |
24 |
15 23
|
eqtrd |
|- ( ( ph /\ X = .0. ) -> ( ( N ` { X } ) i^i ( N ` { Y } ) ) = { .0. } ) |
25 |
4
|
adantr |
|- ( ( ph /\ X =/= .0. ) -> W e. LVec ) |
26 |
18
|
adantr |
|- ( ( ph /\ X =/= .0. ) -> ( N ` { Y } ) e. ( LSubSp ` W ) ) |
27 |
5
|
adantr |
|- ( ( ph /\ X =/= .0. ) -> X e. V ) |
28 |
7
|
adantr |
|- ( ( ph /\ X =/= .0. ) -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
29 |
25
|
adantr |
|- ( ( ( ph /\ X =/= .0. ) /\ X e. ( N ` { Y } ) ) -> W e. LVec ) |
30 |
6
|
adantr |
|- ( ( ph /\ X =/= .0. ) -> Y e. V ) |
31 |
30
|
adantr |
|- ( ( ( ph /\ X =/= .0. ) /\ X e. ( N ` { Y } ) ) -> Y e. V ) |
32 |
|
simpr |
|- ( ( ( ph /\ X =/= .0. ) /\ X e. ( N ` { Y } ) ) -> X e. ( N ` { Y } ) ) |
33 |
|
simplr |
|- ( ( ( ph /\ X =/= .0. ) /\ X e. ( N ` { Y } ) ) -> X =/= .0. ) |
34 |
1 2 3 29 31 32 33
|
lspsneleq |
|- ( ( ( ph /\ X =/= .0. ) /\ X e. ( N ` { Y } ) ) -> ( N ` { X } ) = ( N ` { Y } ) ) |
35 |
34
|
ex |
|- ( ( ph /\ X =/= .0. ) -> ( X e. ( N ` { Y } ) -> ( N ` { X } ) = ( N ` { Y } ) ) ) |
36 |
35
|
necon3ad |
|- ( ( ph /\ X =/= .0. ) -> ( ( N ` { X } ) =/= ( N ` { Y } ) -> -. X e. ( N ` { Y } ) ) ) |
37 |
28 36
|
mpd |
|- ( ( ph /\ X =/= .0. ) -> -. X e. ( N ` { Y } ) ) |
38 |
1 2 3 16 25 26 27 37
|
lspdisj |
|- ( ( ph /\ X =/= .0. ) -> ( ( N ` { X } ) i^i ( N ` { Y } ) ) = { .0. } ) |
39 |
24 38
|
pm2.61dane |
|- ( ph -> ( ( N ` { X } ) i^i ( N ` { Y } ) ) = { .0. } ) |