Step |
Hyp |
Ref |
Expression |
1 |
|
lsmcntz.p |
|- .(+) = ( LSSum ` G ) |
2 |
|
lsmcntz.s |
|- ( ph -> S e. ( SubGrp ` G ) ) |
3 |
|
lsmcntz.t |
|- ( ph -> T e. ( SubGrp ` G ) ) |
4 |
|
lsmcntz.u |
|- ( ph -> U e. ( SubGrp ` G ) ) |
5 |
|
lsmdisj.o |
|- .0. = ( 0g ` G ) |
6 |
|
lsmdisj.i |
|- ( ph -> ( ( S .(+) T ) i^i U ) = { .0. } ) |
7 |
|
lsmdisj2.i |
|- ( ph -> ( S i^i T ) = { .0. } ) |
8 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
9 |
8 1
|
lsmelval |
|- ( ( S e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( x e. ( S .(+) U ) <-> E. s e. S E. u e. U x = ( s ( +g ` G ) u ) ) ) |
10 |
2 4 9
|
syl2anc |
|- ( ph -> ( x e. ( S .(+) U ) <-> E. s e. S E. u e. U x = ( s ( +g ` G ) u ) ) ) |
11 |
|
simplrl |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> s e. S ) |
12 |
|
subgrcl |
|- ( S e. ( SubGrp ` G ) -> G e. Grp ) |
13 |
2 12
|
syl |
|- ( ph -> G e. Grp ) |
14 |
13
|
ad2antrr |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> G e. Grp ) |
15 |
2
|
ad2antrr |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> S e. ( SubGrp ` G ) ) |
16 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
17 |
16
|
subgss |
|- ( S e. ( SubGrp ` G ) -> S C_ ( Base ` G ) ) |
18 |
15 17
|
syl |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> S C_ ( Base ` G ) ) |
19 |
18 11
|
sseldd |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> s e. ( Base ` G ) ) |
20 |
|
eqid |
|- ( invg ` G ) = ( invg ` G ) |
21 |
16 8 5 20
|
grplinv |
|- ( ( G e. Grp /\ s e. ( Base ` G ) ) -> ( ( ( invg ` G ) ` s ) ( +g ` G ) s ) = .0. ) |
22 |
14 19 21
|
syl2anc |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( ( ( invg ` G ) ` s ) ( +g ` G ) s ) = .0. ) |
23 |
22
|
oveq1d |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( ( ( ( invg ` G ) ` s ) ( +g ` G ) s ) ( +g ` G ) u ) = ( .0. ( +g ` G ) u ) ) |
24 |
20
|
subginvcl |
|- ( ( S e. ( SubGrp ` G ) /\ s e. S ) -> ( ( invg ` G ) ` s ) e. S ) |
25 |
15 11 24
|
syl2anc |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( ( invg ` G ) ` s ) e. S ) |
26 |
18 25
|
sseldd |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( ( invg ` G ) ` s ) e. ( Base ` G ) ) |
27 |
4
|
ad2antrr |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> U e. ( SubGrp ` G ) ) |
28 |
16
|
subgss |
|- ( U e. ( SubGrp ` G ) -> U C_ ( Base ` G ) ) |
29 |
27 28
|
syl |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> U C_ ( Base ` G ) ) |
30 |
|
simplrr |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> u e. U ) |
31 |
29 30
|
sseldd |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> u e. ( Base ` G ) ) |
32 |
16 8
|
grpass |
|- ( ( G e. Grp /\ ( ( ( invg ` G ) ` s ) e. ( Base ` G ) /\ s e. ( Base ` G ) /\ u e. ( Base ` G ) ) ) -> ( ( ( ( invg ` G ) ` s ) ( +g ` G ) s ) ( +g ` G ) u ) = ( ( ( invg ` G ) ` s ) ( +g ` G ) ( s ( +g ` G ) u ) ) ) |
33 |
14 26 19 31 32
|
syl13anc |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( ( ( ( invg ` G ) ` s ) ( +g ` G ) s ) ( +g ` G ) u ) = ( ( ( invg ` G ) ` s ) ( +g ` G ) ( s ( +g ` G ) u ) ) ) |
34 |
16 8 5
|
grplid |
|- ( ( G e. Grp /\ u e. ( Base ` G ) ) -> ( .0. ( +g ` G ) u ) = u ) |
35 |
14 31 34
|
syl2anc |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( .0. ( +g ` G ) u ) = u ) |
36 |
23 33 35
|
3eqtr3d |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( ( ( invg ` G ) ` s ) ( +g ` G ) ( s ( +g ` G ) u ) ) = u ) |
37 |
3
|
ad2antrr |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> T e. ( SubGrp ` G ) ) |
38 |
|
simpr |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( s ( +g ` G ) u ) e. T ) |
39 |
8 1
|
lsmelvali |
|- ( ( ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) ) /\ ( ( ( invg ` G ) ` s ) e. S /\ ( s ( +g ` G ) u ) e. T ) ) -> ( ( ( invg ` G ) ` s ) ( +g ` G ) ( s ( +g ` G ) u ) ) e. ( S .(+) T ) ) |
40 |
15 37 25 38 39
|
syl22anc |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( ( ( invg ` G ) ` s ) ( +g ` G ) ( s ( +g ` G ) u ) ) e. ( S .(+) T ) ) |
41 |
36 40
|
eqeltrrd |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> u e. ( S .(+) T ) ) |
42 |
41 30
|
elind |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> u e. ( ( S .(+) T ) i^i U ) ) |
43 |
6
|
ad2antrr |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( ( S .(+) T ) i^i U ) = { .0. } ) |
44 |
42 43
|
eleqtrd |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> u e. { .0. } ) |
45 |
|
elsni |
|- ( u e. { .0. } -> u = .0. ) |
46 |
44 45
|
syl |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> u = .0. ) |
47 |
46
|
oveq2d |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( s ( +g ` G ) u ) = ( s ( +g ` G ) .0. ) ) |
48 |
16 8 5
|
grprid |
|- ( ( G e. Grp /\ s e. ( Base ` G ) ) -> ( s ( +g ` G ) .0. ) = s ) |
49 |
14 19 48
|
syl2anc |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( s ( +g ` G ) .0. ) = s ) |
50 |
47 49
|
eqtrd |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( s ( +g ` G ) u ) = s ) |
51 |
50 38
|
eqeltrrd |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> s e. T ) |
52 |
11 51
|
elind |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> s e. ( S i^i T ) ) |
53 |
7
|
ad2antrr |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( S i^i T ) = { .0. } ) |
54 |
52 53
|
eleqtrd |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> s e. { .0. } ) |
55 |
|
elsni |
|- ( s e. { .0. } -> s = .0. ) |
56 |
54 55
|
syl |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> s = .0. ) |
57 |
56 46
|
oveq12d |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( s ( +g ` G ) u ) = ( .0. ( +g ` G ) .0. ) ) |
58 |
16 5
|
grpidcl |
|- ( G e. Grp -> .0. e. ( Base ` G ) ) |
59 |
16 8 5
|
grplid |
|- ( ( G e. Grp /\ .0. e. ( Base ` G ) ) -> ( .0. ( +g ` G ) .0. ) = .0. ) |
60 |
13 58 59
|
syl2anc2 |
|- ( ph -> ( .0. ( +g ` G ) .0. ) = .0. ) |
61 |
60
|
ad2antrr |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( .0. ( +g ` G ) .0. ) = .0. ) |
62 |
57 61
|
eqtrd |
|- ( ( ( ph /\ ( s e. S /\ u e. U ) ) /\ ( s ( +g ` G ) u ) e. T ) -> ( s ( +g ` G ) u ) = .0. ) |
63 |
62
|
ex |
|- ( ( ph /\ ( s e. S /\ u e. U ) ) -> ( ( s ( +g ` G ) u ) e. T -> ( s ( +g ` G ) u ) = .0. ) ) |
64 |
|
eleq1 |
|- ( x = ( s ( +g ` G ) u ) -> ( x e. T <-> ( s ( +g ` G ) u ) e. T ) ) |
65 |
|
eqeq1 |
|- ( x = ( s ( +g ` G ) u ) -> ( x = .0. <-> ( s ( +g ` G ) u ) = .0. ) ) |
66 |
64 65
|
imbi12d |
|- ( x = ( s ( +g ` G ) u ) -> ( ( x e. T -> x = .0. ) <-> ( ( s ( +g ` G ) u ) e. T -> ( s ( +g ` G ) u ) = .0. ) ) ) |
67 |
63 66
|
syl5ibrcom |
|- ( ( ph /\ ( s e. S /\ u e. U ) ) -> ( x = ( s ( +g ` G ) u ) -> ( x e. T -> x = .0. ) ) ) |
68 |
67
|
rexlimdvva |
|- ( ph -> ( E. s e. S E. u e. U x = ( s ( +g ` G ) u ) -> ( x e. T -> x = .0. ) ) ) |
69 |
10 68
|
sylbid |
|- ( ph -> ( x e. ( S .(+) U ) -> ( x e. T -> x = .0. ) ) ) |
70 |
69
|
impcomd |
|- ( ph -> ( ( x e. T /\ x e. ( S .(+) U ) ) -> x = .0. ) ) |
71 |
|
elin |
|- ( x e. ( T i^i ( S .(+) U ) ) <-> ( x e. T /\ x e. ( S .(+) U ) ) ) |
72 |
|
velsn |
|- ( x e. { .0. } <-> x = .0. ) |
73 |
70 71 72
|
3imtr4g |
|- ( ph -> ( x e. ( T i^i ( S .(+) U ) ) -> x e. { .0. } ) ) |
74 |
73
|
ssrdv |
|- ( ph -> ( T i^i ( S .(+) U ) ) C_ { .0. } ) |
75 |
5
|
subg0cl |
|- ( T e. ( SubGrp ` G ) -> .0. e. T ) |
76 |
3 75
|
syl |
|- ( ph -> .0. e. T ) |
77 |
1
|
lsmub1 |
|- ( ( S e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> S C_ ( S .(+) U ) ) |
78 |
2 4 77
|
syl2anc |
|- ( ph -> S C_ ( S .(+) U ) ) |
79 |
5
|
subg0cl |
|- ( S e. ( SubGrp ` G ) -> .0. e. S ) |
80 |
2 79
|
syl |
|- ( ph -> .0. e. S ) |
81 |
78 80
|
sseldd |
|- ( ph -> .0. e. ( S .(+) U ) ) |
82 |
76 81
|
elind |
|- ( ph -> .0. e. ( T i^i ( S .(+) U ) ) ) |
83 |
82
|
snssd |
|- ( ph -> { .0. } C_ ( T i^i ( S .(+) U ) ) ) |
84 |
74 83
|
eqssd |
|- ( ph -> ( T i^i ( S .(+) U ) ) = { .0. } ) |