Step |
Hyp |
Ref |
Expression |
1 |
|
lsatcmp.a |
|- A = ( LSAtoms ` W ) |
2 |
|
lsatcmp.w |
|- ( ph -> W e. LVec ) |
3 |
|
lsatcmp.t |
|- ( ph -> T e. A ) |
4 |
|
lsatcmp.u |
|- ( ph -> U e. A ) |
5 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
6 |
2 5
|
syl |
|- ( ph -> W e. LMod ) |
7 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
8 |
|
eqid |
|- ( LSpan ` W ) = ( LSpan ` W ) |
9 |
|
eqid |
|- ( 0g ` W ) = ( 0g ` W ) |
10 |
7 8 9 1
|
islsat |
|- ( W e. LMod -> ( U e. A <-> E. v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) U = ( ( LSpan ` W ) ` { v } ) ) ) |
11 |
6 10
|
syl |
|- ( ph -> ( U e. A <-> E. v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) U = ( ( LSpan ` W ) ` { v } ) ) ) |
12 |
4 11
|
mpbid |
|- ( ph -> E. v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) U = ( ( LSpan ` W ) ` { v } ) ) |
13 |
|
eldifsn |
|- ( v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) <-> ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) |
14 |
9 1 6 3
|
lsatn0 |
|- ( ph -> T =/= { ( 0g ` W ) } ) |
15 |
14
|
ad2antrr |
|- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> T =/= { ( 0g ` W ) } ) |
16 |
2
|
ad2antrr |
|- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> W e. LVec ) |
17 |
|
eqid |
|- ( LSubSp ` W ) = ( LSubSp ` W ) |
18 |
17 1 6 3
|
lsatlssel |
|- ( ph -> T e. ( LSubSp ` W ) ) |
19 |
18
|
ad2antrr |
|- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> T e. ( LSubSp ` W ) ) |
20 |
|
simplrl |
|- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> v e. ( Base ` W ) ) |
21 |
|
simpr |
|- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> T C_ ( ( LSpan ` W ) ` { v } ) ) |
22 |
7 9 17 8
|
lspsnat |
|- ( ( ( W e. LVec /\ T e. ( LSubSp ` W ) /\ v e. ( Base ` W ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> ( T = ( ( LSpan ` W ) ` { v } ) \/ T = { ( 0g ` W ) } ) ) |
23 |
16 19 20 21 22
|
syl31anc |
|- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> ( T = ( ( LSpan ` W ) ` { v } ) \/ T = { ( 0g ` W ) } ) ) |
24 |
23
|
ord |
|- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> ( -. T = ( ( LSpan ` W ) ` { v } ) -> T = { ( 0g ` W ) } ) ) |
25 |
24
|
necon1ad |
|- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> ( T =/= { ( 0g ` W ) } -> T = ( ( LSpan ` W ) ` { v } ) ) ) |
26 |
15 25
|
mpd |
|- ( ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) /\ T C_ ( ( LSpan ` W ) ` { v } ) ) -> T = ( ( LSpan ` W ) ` { v } ) ) |
27 |
26
|
ex |
|- ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) -> ( T C_ ( ( LSpan ` W ) ` { v } ) -> T = ( ( LSpan ` W ) ` { v } ) ) ) |
28 |
|
eqimss |
|- ( T = ( ( LSpan ` W ) ` { v } ) -> T C_ ( ( LSpan ` W ) ` { v } ) ) |
29 |
27 28
|
impbid1 |
|- ( ( ph /\ ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) ) -> ( T C_ ( ( LSpan ` W ) ` { v } ) <-> T = ( ( LSpan ` W ) ` { v } ) ) ) |
30 |
29
|
ex |
|- ( ph -> ( ( v e. ( Base ` W ) /\ v =/= ( 0g ` W ) ) -> ( T C_ ( ( LSpan ` W ) ` { v } ) <-> T = ( ( LSpan ` W ) ` { v } ) ) ) ) |
31 |
13 30
|
syl5bi |
|- ( ph -> ( v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) -> ( T C_ ( ( LSpan ` W ) ` { v } ) <-> T = ( ( LSpan ` W ) ` { v } ) ) ) ) |
32 |
|
sseq2 |
|- ( U = ( ( LSpan ` W ) ` { v } ) -> ( T C_ U <-> T C_ ( ( LSpan ` W ) ` { v } ) ) ) |
33 |
|
eqeq2 |
|- ( U = ( ( LSpan ` W ) ` { v } ) -> ( T = U <-> T = ( ( LSpan ` W ) ` { v } ) ) ) |
34 |
32 33
|
bibi12d |
|- ( U = ( ( LSpan ` W ) ` { v } ) -> ( ( T C_ U <-> T = U ) <-> ( T C_ ( ( LSpan ` W ) ` { v } ) <-> T = ( ( LSpan ` W ) ` { v } ) ) ) ) |
35 |
34
|
biimprcd |
|- ( ( T C_ ( ( LSpan ` W ) ` { v } ) <-> T = ( ( LSpan ` W ) ` { v } ) ) -> ( U = ( ( LSpan ` W ) ` { v } ) -> ( T C_ U <-> T = U ) ) ) |
36 |
31 35
|
syl6 |
|- ( ph -> ( v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) -> ( U = ( ( LSpan ` W ) ` { v } ) -> ( T C_ U <-> T = U ) ) ) ) |
37 |
36
|
rexlimdv |
|- ( ph -> ( E. v e. ( ( Base ` W ) \ { ( 0g ` W ) } ) U = ( ( LSpan ` W ) ` { v } ) -> ( T C_ U <-> T = U ) ) ) |
38 |
12 37
|
mpd |
|- ( ph -> ( T C_ U <-> T = U ) ) |