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 |
1
|
lsmub1 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑆 ⊆ ( 𝑆 ⊕ 𝑇 ) ) |
8 |
2 3 7
|
syl2anc |
⊢ ( 𝜑 → 𝑆 ⊆ ( 𝑆 ⊕ 𝑇 ) ) |
9 |
8
|
ssrind |
⊢ ( 𝜑 → ( 𝑆 ∩ 𝑈 ) ⊆ ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) ) |
10 |
9 6
|
sseqtrd |
⊢ ( 𝜑 → ( 𝑆 ∩ 𝑈 ) ⊆ { 0 } ) |
11 |
5
|
subg0cl |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 0 ∈ 𝑆 ) |
12 |
2 11
|
syl |
⊢ ( 𝜑 → 0 ∈ 𝑆 ) |
13 |
5
|
subg0cl |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 0 ∈ 𝑈 ) |
14 |
4 13
|
syl |
⊢ ( 𝜑 → 0 ∈ 𝑈 ) |
15 |
12 14
|
elind |
⊢ ( 𝜑 → 0 ∈ ( 𝑆 ∩ 𝑈 ) ) |
16 |
15
|
snssd |
⊢ ( 𝜑 → { 0 } ⊆ ( 𝑆 ∩ 𝑈 ) ) |
17 |
10 16
|
eqssd |
⊢ ( 𝜑 → ( 𝑆 ∩ 𝑈 ) = { 0 } ) |
18 |
1
|
lsmub2 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑇 ⊆ ( 𝑆 ⊕ 𝑇 ) ) |
19 |
2 3 18
|
syl2anc |
⊢ ( 𝜑 → 𝑇 ⊆ ( 𝑆 ⊕ 𝑇 ) ) |
20 |
19
|
ssrind |
⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) ⊆ ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) ) |
21 |
20 6
|
sseqtrd |
⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) ⊆ { 0 } ) |
22 |
5
|
subg0cl |
⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 0 ∈ 𝑇 ) |
23 |
3 22
|
syl |
⊢ ( 𝜑 → 0 ∈ 𝑇 ) |
24 |
23 14
|
elind |
⊢ ( 𝜑 → 0 ∈ ( 𝑇 ∩ 𝑈 ) ) |
25 |
24
|
snssd |
⊢ ( 𝜑 → { 0 } ⊆ ( 𝑇 ∩ 𝑈 ) ) |
26 |
21 25
|
eqssd |
⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) = { 0 } ) |
27 |
17 26
|
jca |
⊢ ( 𝜑 → ( ( 𝑆 ∩ 𝑈 ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) ) |