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 |
|
lsmdisj3.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
9 |
|
lsmdisj3.s |
⊢ ( 𝜑 → 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) ) |
10 |
1 8
|
lsmcom2 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) ) → ( 𝑆 ⊕ 𝑇 ) = ( 𝑇 ⊕ 𝑆 ) ) |
11 |
2 3 9 10
|
syl3anc |
⊢ ( 𝜑 → ( 𝑆 ⊕ 𝑇 ) = ( 𝑇 ⊕ 𝑆 ) ) |
12 |
11
|
ineq1d |
⊢ ( 𝜑 → ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = ( ( 𝑇 ⊕ 𝑆 ) ∩ 𝑈 ) ) |
13 |
12 6
|
eqtr3d |
⊢ ( 𝜑 → ( ( 𝑇 ⊕ 𝑆 ) ∩ 𝑈 ) = { 0 } ) |
14 |
|
incom |
⊢ ( 𝑇 ∩ 𝑆 ) = ( 𝑆 ∩ 𝑇 ) |
15 |
14 7
|
eqtrid |
⊢ ( 𝜑 → ( 𝑇 ∩ 𝑆 ) = { 0 } ) |
16 |
1 3 2 4 5 13 15
|
lsmdisj2 |
⊢ ( 𝜑 → ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } ) |