Step |
Hyp |
Ref |
Expression |
1 |
|
lcvat.s |
|- S = ( LSubSp ` W ) |
2 |
|
lcvat.p |
|- .(+) = ( LSSum ` W ) |
3 |
|
lcvat.a |
|- A = ( LSAtoms ` W ) |
4 |
|
icvat.c |
|- C = (
|
5 |
|
lcvat.w |
|- ( ph -> W e. LMod ) |
6 |
|
lcvat.t |
|- ( ph -> T e. S ) |
7 |
|
lcvat.u |
|- ( ph -> U e. S ) |
8 |
|
lcvat.l |
|- ( ph -> T C U ) |
9 |
1 4 5 6 7 8
|
lcvpss |
|- ( ph -> T C. U ) |
10 |
1 2 3 5 6 7 9
|
lrelat |
|- ( ph -> E. q e. A ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) ) |
11 |
5
|
3ad2ant1 |
|- ( ( ph /\ q e. A /\ ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) ) -> W e. LMod ) |
12 |
6
|
3ad2ant1 |
|- ( ( ph /\ q e. A /\ ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) ) -> T e. S ) |
13 |
7
|
3ad2ant1 |
|- ( ( ph /\ q e. A /\ ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) ) -> U e. S ) |
14 |
|
simp2 |
|- ( ( ph /\ q e. A /\ ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) ) -> q e. A ) |
15 |
1 3 11 14
|
lsatlssel |
|- ( ( ph /\ q e. A /\ ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) ) -> q e. S ) |
16 |
1 2
|
lsmcl |
|- ( ( W e. LMod /\ T e. S /\ q e. S ) -> ( T .(+) q ) e. S ) |
17 |
11 12 15 16
|
syl3anc |
|- ( ( ph /\ q e. A /\ ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) ) -> ( T .(+) q ) e. S ) |
18 |
8
|
3ad2ant1 |
|- ( ( ph /\ q e. A /\ ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) ) -> T C U ) |
19 |
|
simp3l |
|- ( ( ph /\ q e. A /\ ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) ) -> T C. ( T .(+) q ) ) |
20 |
|
simp3r |
|- ( ( ph /\ q e. A /\ ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) ) -> ( T .(+) q ) C_ U ) |
21 |
1 4 11 12 13 17 18 19 20
|
lcvnbtwn2 |
|- ( ( ph /\ q e. A /\ ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) ) -> ( T .(+) q ) = U ) |
22 |
21
|
3exp |
|- ( ph -> ( q e. A -> ( ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) -> ( T .(+) q ) = U ) ) ) |
23 |
22
|
reximdvai |
|- ( ph -> ( E. q e. A ( T C. ( T .(+) q ) /\ ( T .(+) q ) C_ U ) -> E. q e. A ( T .(+) q ) = U ) ) |
24 |
10 23
|
mpd |
|- ( ph -> E. q e. A ( T .(+) q ) = U ) |