Step |
Hyp |
Ref |
Expression |
1 |
|
lsateln0.z |
|- .0. = ( 0g ` W ) |
2 |
|
lsateln0.a |
|- A = ( LSAtoms ` W ) |
3 |
|
lsateln0.w |
|- ( ph -> W e. LMod ) |
4 |
|
lsateln0.u |
|- ( ph -> U e. A ) |
5 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
6 |
|
eqid |
|- ( LSpan ` W ) = ( LSpan ` W ) |
7 |
5 6 1 2
|
islsat |
|- ( W e. LMod -> ( U e. A <-> E. v e. ( ( Base ` W ) \ { .0. } ) U = ( ( LSpan ` W ) ` { v } ) ) ) |
8 |
3 7
|
syl |
|- ( ph -> ( U e. A <-> E. v e. ( ( Base ` W ) \ { .0. } ) U = ( ( LSpan ` W ) ` { v } ) ) ) |
9 |
4 8
|
mpbid |
|- ( ph -> E. v e. ( ( Base ` W ) \ { .0. } ) U = ( ( LSpan ` W ) ` { v } ) ) |
10 |
|
eldifi |
|- ( v e. ( ( Base ` W ) \ { .0. } ) -> v e. ( Base ` W ) ) |
11 |
5 6
|
lspsnid |
|- ( ( W e. LMod /\ v e. ( Base ` W ) ) -> v e. ( ( LSpan ` W ) ` { v } ) ) |
12 |
3 10 11
|
syl2an |
|- ( ( ph /\ v e. ( ( Base ` W ) \ { .0. } ) ) -> v e. ( ( LSpan ` W ) ` { v } ) ) |
13 |
|
eleq2 |
|- ( U = ( ( LSpan ` W ) ` { v } ) -> ( v e. U <-> v e. ( ( LSpan ` W ) ` { v } ) ) ) |
14 |
12 13
|
syl5ibrcom |
|- ( ( ph /\ v e. ( ( Base ` W ) \ { .0. } ) ) -> ( U = ( ( LSpan ` W ) ` { v } ) -> v e. U ) ) |
15 |
14
|
reximdva |
|- ( ph -> ( E. v e. ( ( Base ` W ) \ { .0. } ) U = ( ( LSpan ` W ) ` { v } ) -> E. v e. ( ( Base ` W ) \ { .0. } ) v e. U ) ) |
16 |
9 15
|
mpd |
|- ( ph -> E. v e. ( ( Base ` W ) \ { .0. } ) v e. U ) |
17 |
|
eldifsn |
|- ( v e. ( ( Base ` W ) \ { .0. } ) <-> ( v e. ( Base ` W ) /\ v =/= .0. ) ) |
18 |
17
|
anbi1i |
|- ( ( v e. ( ( Base ` W ) \ { .0. } ) /\ v e. U ) <-> ( ( v e. ( Base ` W ) /\ v =/= .0. ) /\ v e. U ) ) |
19 |
|
anass |
|- ( ( ( v e. ( Base ` W ) /\ v =/= .0. ) /\ v e. U ) <-> ( v e. ( Base ` W ) /\ ( v =/= .0. /\ v e. U ) ) ) |
20 |
18 19
|
bitri |
|- ( ( v e. ( ( Base ` W ) \ { .0. } ) /\ v e. U ) <-> ( v e. ( Base ` W ) /\ ( v =/= .0. /\ v e. U ) ) ) |
21 |
20
|
simprbi |
|- ( ( v e. ( ( Base ` W ) \ { .0. } ) /\ v e. U ) -> ( v =/= .0. /\ v e. U ) ) |
22 |
21
|
ancomd |
|- ( ( v e. ( ( Base ` W ) \ { .0. } ) /\ v e. U ) -> ( v e. U /\ v =/= .0. ) ) |
23 |
22
|
reximi2 |
|- ( E. v e. ( ( Base ` W ) \ { .0. } ) v e. U -> E. v e. U v =/= .0. ) |
24 |
16 23
|
syl |
|- ( ph -> E. v e. U v =/= .0. ) |