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