Step |
Hyp |
Ref |
Expression |
1 |
|
lshpnelb.v |
|- V = ( Base ` W ) |
2 |
|
lshpnelb.n |
|- N = ( LSpan ` W ) |
3 |
|
lshpnelb.p |
|- .(+) = ( LSSum ` W ) |
4 |
|
lshpnelb.h |
|- H = ( LSHyp ` W ) |
5 |
|
lshpnelb.w |
|- ( ph -> W e. LVec ) |
6 |
|
lshpnelb.u |
|- ( ph -> U e. H ) |
7 |
|
lshpnelb.x |
|- ( ph -> X e. V ) |
8 |
|
eqid |
|- ( LSubSp ` W ) = ( LSubSp ` W ) |
9 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
10 |
5 9
|
syl |
|- ( ph -> W e. LMod ) |
11 |
1 2 8 3 4 10
|
islshpsm |
|- ( ph -> ( U e. H <-> ( U e. ( LSubSp ` W ) /\ U =/= V /\ E. v e. V ( U .(+) ( N ` { v } ) ) = V ) ) ) |
12 |
6 11
|
mpbid |
|- ( ph -> ( U e. ( LSubSp ` W ) /\ U =/= V /\ E. v e. V ( U .(+) ( N ` { v } ) ) = V ) ) |
13 |
12
|
simp3d |
|- ( ph -> E. v e. V ( U .(+) ( N ` { v } ) ) = V ) |
14 |
13
|
adantr |
|- ( ( ph /\ -. X e. U ) -> E. v e. V ( U .(+) ( N ` { v } ) ) = V ) |
15 |
|
simp1l |
|- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> ph ) |
16 |
|
simp2 |
|- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> v e. V ) |
17 |
8
|
lsssssubg |
|- ( W e. LMod -> ( LSubSp ` W ) C_ ( SubGrp ` W ) ) |
18 |
10 17
|
syl |
|- ( ph -> ( LSubSp ` W ) C_ ( SubGrp ` W ) ) |
19 |
8 4 10 6
|
lshplss |
|- ( ph -> U e. ( LSubSp ` W ) ) |
20 |
18 19
|
sseldd |
|- ( ph -> U e. ( SubGrp ` W ) ) |
21 |
1 8 2
|
lspsncl |
|- ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` W ) ) |
22 |
10 7 21
|
syl2anc |
|- ( ph -> ( N ` { X } ) e. ( LSubSp ` W ) ) |
23 |
18 22
|
sseldd |
|- ( ph -> ( N ` { X } ) e. ( SubGrp ` W ) ) |
24 |
3
|
lsmub1 |
|- ( ( U e. ( SubGrp ` W ) /\ ( N ` { X } ) e. ( SubGrp ` W ) ) -> U C_ ( U .(+) ( N ` { X } ) ) ) |
25 |
20 23 24
|
syl2anc |
|- ( ph -> U C_ ( U .(+) ( N ` { X } ) ) ) |
26 |
25
|
adantr |
|- ( ( ph /\ -. X e. U ) -> U C_ ( U .(+) ( N ` { X } ) ) ) |
27 |
3
|
lsmub2 |
|- ( ( U e. ( SubGrp ` W ) /\ ( N ` { X } ) e. ( SubGrp ` W ) ) -> ( N ` { X } ) C_ ( U .(+) ( N ` { X } ) ) ) |
28 |
20 23 27
|
syl2anc |
|- ( ph -> ( N ` { X } ) C_ ( U .(+) ( N ` { X } ) ) ) |
29 |
1 2
|
lspsnid |
|- ( ( W e. LMod /\ X e. V ) -> X e. ( N ` { X } ) ) |
30 |
10 7 29
|
syl2anc |
|- ( ph -> X e. ( N ` { X } ) ) |
31 |
28 30
|
sseldd |
|- ( ph -> X e. ( U .(+) ( N ` { X } ) ) ) |
32 |
|
nelne1 |
|- ( ( X e. ( U .(+) ( N ` { X } ) ) /\ -. X e. U ) -> ( U .(+) ( N ` { X } ) ) =/= U ) |
33 |
31 32
|
sylan |
|- ( ( ph /\ -. X e. U ) -> ( U .(+) ( N ` { X } ) ) =/= U ) |
34 |
33
|
necomd |
|- ( ( ph /\ -. X e. U ) -> U =/= ( U .(+) ( N ` { X } ) ) ) |
35 |
|
df-pss |
|- ( U C. ( U .(+) ( N ` { X } ) ) <-> ( U C_ ( U .(+) ( N ` { X } ) ) /\ U =/= ( U .(+) ( N ` { X } ) ) ) ) |
36 |
26 34 35
|
sylanbrc |
|- ( ( ph /\ -. X e. U ) -> U C. ( U .(+) ( N ` { X } ) ) ) |
37 |
36
|
3ad2ant1 |
|- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> U C. ( U .(+) ( N ` { X } ) ) ) |
38 |
8 3
|
lsmcl |
|- ( ( W e. LMod /\ U e. ( LSubSp ` W ) /\ ( N ` { X } ) e. ( LSubSp ` W ) ) -> ( U .(+) ( N ` { X } ) ) e. ( LSubSp ` W ) ) |
39 |
10 19 22 38
|
syl3anc |
|- ( ph -> ( U .(+) ( N ` { X } ) ) e. ( LSubSp ` W ) ) |
40 |
1 8
|
lssss |
|- ( ( U .(+) ( N ` { X } ) ) e. ( LSubSp ` W ) -> ( U .(+) ( N ` { X } ) ) C_ V ) |
41 |
39 40
|
syl |
|- ( ph -> ( U .(+) ( N ` { X } ) ) C_ V ) |
42 |
41
|
adantr |
|- ( ( ph /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) C_ V ) |
43 |
|
simpr |
|- ( ( ph /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { v } ) ) = V ) |
44 |
42 43
|
sseqtrrd |
|- ( ( ph /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) C_ ( U .(+) ( N ` { v } ) ) ) |
45 |
44
|
adantlr |
|- ( ( ( ph /\ -. X e. U ) /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) C_ ( U .(+) ( N ` { v } ) ) ) |
46 |
45
|
3adant2 |
|- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) C_ ( U .(+) ( N ` { v } ) ) ) |
47 |
5
|
adantr |
|- ( ( ph /\ v e. V ) -> W e. LVec ) |
48 |
19
|
adantr |
|- ( ( ph /\ v e. V ) -> U e. ( LSubSp ` W ) ) |
49 |
39
|
adantr |
|- ( ( ph /\ v e. V ) -> ( U .(+) ( N ` { X } ) ) e. ( LSubSp ` W ) ) |
50 |
|
simpr |
|- ( ( ph /\ v e. V ) -> v e. V ) |
51 |
1 8 2 3 47 48 49 50
|
lsmcv |
|- ( ( ( ph /\ v e. V ) /\ U C. ( U .(+) ( N ` { X } ) ) /\ ( U .(+) ( N ` { X } ) ) C_ ( U .(+) ( N ` { v } ) ) ) -> ( U .(+) ( N ` { X } ) ) = ( U .(+) ( N ` { v } ) ) ) |
52 |
15 16 37 46 51
|
syl211anc |
|- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) = ( U .(+) ( N ` { v } ) ) ) |
53 |
|
simp3 |
|- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { v } ) ) = V ) |
54 |
52 53
|
eqtrd |
|- ( ( ( ph /\ -. X e. U ) /\ v e. V /\ ( U .(+) ( N ` { v } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) = V ) |
55 |
54
|
rexlimdv3a |
|- ( ( ph /\ -. X e. U ) -> ( E. v e. V ( U .(+) ( N ` { v } ) ) = V -> ( U .(+) ( N ` { X } ) ) = V ) ) |
56 |
14 55
|
mpd |
|- ( ( ph /\ -. X e. U ) -> ( U .(+) ( N ` { X } ) ) = V ) |
57 |
10
|
adantr |
|- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> W e. LMod ) |
58 |
6
|
adantr |
|- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> U e. H ) |
59 |
7
|
adantr |
|- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> X e. V ) |
60 |
|
simpr |
|- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> ( U .(+) ( N ` { X } ) ) = V ) |
61 |
1 2 3 4 57 58 59 60
|
lshpnel |
|- ( ( ph /\ ( U .(+) ( N ` { X } ) ) = V ) -> -. X e. U ) |
62 |
56 61
|
impbida |
|- ( ph -> ( -. X e. U <-> ( U .(+) ( N ` { X } ) ) = V ) ) |