Step |
Hyp |
Ref |
Expression |
1 |
|
lcv2.s |
|- S = ( LSubSp ` W ) |
2 |
|
lcv2.p |
|- .(+) = ( LSSum ` W ) |
3 |
|
lcv2.a |
|- A = ( LSAtoms ` W ) |
4 |
|
lcv2.c |
|- C = (
|
5 |
|
lcv2.w |
|- ( ph -> W e. LVec ) |
6 |
|
lcv2.u |
|- ( ph -> U e. S ) |
7 |
|
lcv2.q |
|- ( ph -> Q e. A ) |
8 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
9 |
5 8
|
syl |
|- ( ph -> W e. LMod ) |
10 |
1
|
lsssssubg |
|- ( W e. LMod -> S C_ ( SubGrp ` W ) ) |
11 |
9 10
|
syl |
|- ( ph -> S C_ ( SubGrp ` W ) ) |
12 |
11 6
|
sseldd |
|- ( ph -> U e. ( SubGrp ` W ) ) |
13 |
1 3 9 7
|
lsatlssel |
|- ( ph -> Q e. S ) |
14 |
11 13
|
sseldd |
|- ( ph -> Q e. ( SubGrp ` W ) ) |
15 |
2 12 14
|
lssnle |
|- ( ph -> ( -. Q C_ U <-> U C. ( U .(+) Q ) ) ) |
16 |
1 2 3 4 5 6 7
|
lcv1 |
|- ( ph -> ( -. Q C_ U <-> U C ( U .(+) Q ) ) ) |
17 |
15 16
|
bitr3d |
|- ( ph -> ( U C. ( U .(+) Q ) <-> U C ( U .(+) Q ) ) ) |