Step |
Hyp |
Ref |
Expression |
1 |
|
lsatcvat.o |
|- .0. = ( 0g ` W ) |
2 |
|
lsatcvat.s |
|- S = ( LSubSp ` W ) |
3 |
|
lsatcvat.p |
|- .(+) = ( LSSum ` W ) |
4 |
|
lsatcvat.a |
|- A = ( LSAtoms ` W ) |
5 |
|
lsatcvat.w |
|- ( ph -> W e. LVec ) |
6 |
|
lsatcvat.u |
|- ( ph -> U e. S ) |
7 |
|
lsatcvat.q |
|- ( ph -> Q e. A ) |
8 |
|
lsatcvat.r |
|- ( ph -> R e. A ) |
9 |
|
lsatcvat.n |
|- ( ph -> U =/= { .0. } ) |
10 |
|
lsatcvat.l |
|- ( ph -> U C. ( Q .(+) R ) ) |
11 |
5
|
adantr |
|- ( ( ph /\ -. Q C_ U ) -> W e. LVec ) |
12 |
6
|
adantr |
|- ( ( ph /\ -. Q C_ U ) -> U e. S ) |
13 |
7
|
adantr |
|- ( ( ph /\ -. Q C_ U ) -> Q e. A ) |
14 |
8
|
adantr |
|- ( ( ph /\ -. Q C_ U ) -> R e. A ) |
15 |
9
|
adantr |
|- ( ( ph /\ -. Q C_ U ) -> U =/= { .0. } ) |
16 |
10
|
adantr |
|- ( ( ph /\ -. Q C_ U ) -> U C. ( Q .(+) R ) ) |
17 |
|
simpr |
|- ( ( ph /\ -. Q C_ U ) -> -. Q C_ U ) |
18 |
1 2 3 4 11 12 13 14 15 16 17
|
lsatcvatlem |
|- ( ( ph /\ -. Q C_ U ) -> U e. A ) |
19 |
5
|
adantr |
|- ( ( ph /\ -. R C_ U ) -> W e. LVec ) |
20 |
6
|
adantr |
|- ( ( ph /\ -. R C_ U ) -> U e. S ) |
21 |
8
|
adantr |
|- ( ( ph /\ -. R C_ U ) -> R e. A ) |
22 |
7
|
adantr |
|- ( ( ph /\ -. R C_ U ) -> Q e. A ) |
23 |
9
|
adantr |
|- ( ( ph /\ -. R C_ U ) -> U =/= { .0. } ) |
24 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
25 |
5 24
|
syl |
|- ( ph -> W e. LMod ) |
26 |
|
lmodabl |
|- ( W e. LMod -> W e. Abel ) |
27 |
25 26
|
syl |
|- ( ph -> W e. Abel ) |
28 |
2
|
lsssssubg |
|- ( W e. LMod -> S C_ ( SubGrp ` W ) ) |
29 |
25 28
|
syl |
|- ( ph -> S C_ ( SubGrp ` W ) ) |
30 |
2 4 25 7
|
lsatlssel |
|- ( ph -> Q e. S ) |
31 |
29 30
|
sseldd |
|- ( ph -> Q e. ( SubGrp ` W ) ) |
32 |
2 4 25 8
|
lsatlssel |
|- ( ph -> R e. S ) |
33 |
29 32
|
sseldd |
|- ( ph -> R e. ( SubGrp ` W ) ) |
34 |
3
|
lsmcom |
|- ( ( W e. Abel /\ Q e. ( SubGrp ` W ) /\ R e. ( SubGrp ` W ) ) -> ( Q .(+) R ) = ( R .(+) Q ) ) |
35 |
27 31 33 34
|
syl3anc |
|- ( ph -> ( Q .(+) R ) = ( R .(+) Q ) ) |
36 |
35
|
psseq2d |
|- ( ph -> ( U C. ( Q .(+) R ) <-> U C. ( R .(+) Q ) ) ) |
37 |
10 36
|
mpbid |
|- ( ph -> U C. ( R .(+) Q ) ) |
38 |
37
|
adantr |
|- ( ( ph /\ -. R C_ U ) -> U C. ( R .(+) Q ) ) |
39 |
|
simpr |
|- ( ( ph /\ -. R C_ U ) -> -. R C_ U ) |
40 |
1 2 3 4 19 20 21 22 23 38 39
|
lsatcvatlem |
|- ( ( ph /\ -. R C_ U ) -> U e. A ) |
41 |
29 6
|
sseldd |
|- ( ph -> U e. ( SubGrp ` W ) ) |
42 |
3
|
lsmlub |
|- ( ( Q e. ( SubGrp ` W ) /\ R e. ( SubGrp ` W ) /\ U e. ( SubGrp ` W ) ) -> ( ( Q C_ U /\ R C_ U ) <-> ( Q .(+) R ) C_ U ) ) |
43 |
31 33 41 42
|
syl3anc |
|- ( ph -> ( ( Q C_ U /\ R C_ U ) <-> ( Q .(+) R ) C_ U ) ) |
44 |
|
ssnpss |
|- ( ( Q .(+) R ) C_ U -> -. U C. ( Q .(+) R ) ) |
45 |
43 44
|
syl6bi |
|- ( ph -> ( ( Q C_ U /\ R C_ U ) -> -. U C. ( Q .(+) R ) ) ) |
46 |
45
|
con2d |
|- ( ph -> ( U C. ( Q .(+) R ) -> -. ( Q C_ U /\ R C_ U ) ) ) |
47 |
|
ianor |
|- ( -. ( Q C_ U /\ R C_ U ) <-> ( -. Q C_ U \/ -. R C_ U ) ) |
48 |
46 47
|
syl6ib |
|- ( ph -> ( U C. ( Q .(+) R ) -> ( -. Q C_ U \/ -. R C_ U ) ) ) |
49 |
10 48
|
mpd |
|- ( ph -> ( -. Q C_ U \/ -. R C_ U ) ) |
50 |
18 40 49
|
mpjaodan |
|- ( ph -> U e. A ) |