Step |
Hyp |
Ref |
Expression |
1 |
|
lsatcvat3.s |
|- S = ( LSubSp ` W ) |
2 |
|
lsatcvat3.p |
|- .(+) = ( LSSum ` W ) |
3 |
|
lsatcvat3.a |
|- A = ( LSAtoms ` W ) |
4 |
|
lsatcvat3.w |
|- ( ph -> W e. LVec ) |
5 |
|
lsatcvat3.u |
|- ( ph -> U e. S ) |
6 |
|
lsatcvat3.q |
|- ( ph -> Q e. A ) |
7 |
|
lsatcvat3.r |
|- ( ph -> R e. A ) |
8 |
|
lsatcvat3.n |
|- ( ph -> Q =/= R ) |
9 |
|
lsatcvat3.m |
|- ( ph -> -. R C_ U ) |
10 |
|
lsatcvat3.l |
|- ( ph -> Q C_ ( U .(+) R ) ) |
11 |
|
eqid |
|- (
|
12 |
|
lveclmod |
|- ( W e. LVec -> W e. LMod ) |
13 |
4 12
|
syl |
|- ( ph -> W e. LMod ) |
14 |
1 3 13 6
|
lsatlssel |
|- ( ph -> Q e. S ) |
15 |
1 3 13 7
|
lsatlssel |
|- ( ph -> R e. S ) |
16 |
1 2
|
lsmcl |
|- ( ( W e. LMod /\ Q e. S /\ R e. S ) -> ( Q .(+) R ) e. S ) |
17 |
13 14 15 16
|
syl3anc |
|- ( ph -> ( Q .(+) R ) e. S ) |
18 |
1
|
lssincl |
|- ( ( W e. LMod /\ U e. S /\ ( Q .(+) R ) e. S ) -> ( U i^i ( Q .(+) R ) ) e. S ) |
19 |
13 5 17 18
|
syl3anc |
|- ( ph -> ( U i^i ( Q .(+) R ) ) e. S ) |
20 |
1 2 3 11 4 5 7
|
lcv1 |
|- ( ph -> ( -. R C_ U <-> U (
|
21 |
9 20
|
mpbid |
|- ( ph -> U (
|
22 |
|
lmodabl |
|- ( W e. LMod -> W e. Abel ) |
23 |
13 22
|
syl |
|- ( ph -> W e. Abel ) |
24 |
1
|
lsssssubg |
|- ( W e. LMod -> S C_ ( SubGrp ` W ) ) |
25 |
13 24
|
syl |
|- ( ph -> S C_ ( SubGrp ` W ) ) |
26 |
25 14
|
sseldd |
|- ( ph -> Q e. ( SubGrp ` W ) ) |
27 |
25 15
|
sseldd |
|- ( ph -> R e. ( SubGrp ` W ) ) |
28 |
2
|
lsmcom |
|- ( ( W e. Abel /\ Q e. ( SubGrp ` W ) /\ R e. ( SubGrp ` W ) ) -> ( Q .(+) R ) = ( R .(+) Q ) ) |
29 |
23 26 27 28
|
syl3anc |
|- ( ph -> ( Q .(+) R ) = ( R .(+) Q ) ) |
30 |
29
|
oveq2d |
|- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( U .(+) ( R .(+) Q ) ) ) |
31 |
25 5
|
sseldd |
|- ( ph -> U e. ( SubGrp ` W ) ) |
32 |
2
|
lsmass |
|- ( ( U e. ( SubGrp ` W ) /\ R e. ( SubGrp ` W ) /\ Q e. ( SubGrp ` W ) ) -> ( ( U .(+) R ) .(+) Q ) = ( U .(+) ( R .(+) Q ) ) ) |
33 |
31 27 26 32
|
syl3anc |
|- ( ph -> ( ( U .(+) R ) .(+) Q ) = ( U .(+) ( R .(+) Q ) ) ) |
34 |
30 33
|
eqtr4d |
|- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( ( U .(+) R ) .(+) Q ) ) |
35 |
1 2
|
lsmcl |
|- ( ( W e. LMod /\ U e. S /\ R e. S ) -> ( U .(+) R ) e. S ) |
36 |
13 5 15 35
|
syl3anc |
|- ( ph -> ( U .(+) R ) e. S ) |
37 |
25 36
|
sseldd |
|- ( ph -> ( U .(+) R ) e. ( SubGrp ` W ) ) |
38 |
2
|
lsmless2 |
|- ( ( ( U .(+) R ) e. ( SubGrp ` W ) /\ ( U .(+) R ) e. ( SubGrp ` W ) /\ Q C_ ( U .(+) R ) ) -> ( ( U .(+) R ) .(+) Q ) C_ ( ( U .(+) R ) .(+) ( U .(+) R ) ) ) |
39 |
37 37 10 38
|
syl3anc |
|- ( ph -> ( ( U .(+) R ) .(+) Q ) C_ ( ( U .(+) R ) .(+) ( U .(+) R ) ) ) |
40 |
34 39
|
eqsstrd |
|- ( ph -> ( U .(+) ( Q .(+) R ) ) C_ ( ( U .(+) R ) .(+) ( U .(+) R ) ) ) |
41 |
2
|
lsmidm |
|- ( ( U .(+) R ) e. ( SubGrp ` W ) -> ( ( U .(+) R ) .(+) ( U .(+) R ) ) = ( U .(+) R ) ) |
42 |
37 41
|
syl |
|- ( ph -> ( ( U .(+) R ) .(+) ( U .(+) R ) ) = ( U .(+) R ) ) |
43 |
40 42
|
sseqtrd |
|- ( ph -> ( U .(+) ( Q .(+) R ) ) C_ ( U .(+) R ) ) |
44 |
25 17
|
sseldd |
|- ( ph -> ( Q .(+) R ) e. ( SubGrp ` W ) ) |
45 |
2
|
lsmub2 |
|- ( ( Q e. ( SubGrp ` W ) /\ R e. ( SubGrp ` W ) ) -> R C_ ( Q .(+) R ) ) |
46 |
26 27 45
|
syl2anc |
|- ( ph -> R C_ ( Q .(+) R ) ) |
47 |
2
|
lsmless2 |
|- ( ( U e. ( SubGrp ` W ) /\ ( Q .(+) R ) e. ( SubGrp ` W ) /\ R C_ ( Q .(+) R ) ) -> ( U .(+) R ) C_ ( U .(+) ( Q .(+) R ) ) ) |
48 |
31 44 46 47
|
syl3anc |
|- ( ph -> ( U .(+) R ) C_ ( U .(+) ( Q .(+) R ) ) ) |
49 |
43 48
|
eqssd |
|- ( ph -> ( U .(+) ( Q .(+) R ) ) = ( U .(+) R ) ) |
50 |
21 49
|
breqtrrd |
|- ( ph -> U (
|
51 |
1 2 11 13 5 17 50
|
lcvexchlem4 |
|- ( ph -> ( U i^i ( Q .(+) R ) ) (
|
52 |
1 2 3 11 4 19 6 7 8 51
|
lsatcvat2 |
|- ( ph -> ( U i^i ( Q .(+) R ) ) e. A ) |