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 |
|
lsmdisjr.i |
⊢ ( 𝜑 → ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } ) |
7 |
|
incom |
⊢ ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = ( ( 𝑇 ⊕ 𝑈 ) ∩ 𝑆 ) |
8 |
7 6
|
eqtr3id |
⊢ ( 𝜑 → ( ( 𝑇 ⊕ 𝑈 ) ∩ 𝑆 ) = { 0 } ) |
9 |
1 3 4 2 5 8
|
lsmdisj |
⊢ ( 𝜑 → ( ( 𝑇 ∩ 𝑆 ) = { 0 } ∧ ( 𝑈 ∩ 𝑆 ) = { 0 } ) ) |
10 |
|
incom |
⊢ ( 𝑇 ∩ 𝑆 ) = ( 𝑆 ∩ 𝑇 ) |
11 |
10
|
eqeq1i |
⊢ ( ( 𝑇 ∩ 𝑆 ) = { 0 } ↔ ( 𝑆 ∩ 𝑇 ) = { 0 } ) |
12 |
|
incom |
⊢ ( 𝑈 ∩ 𝑆 ) = ( 𝑆 ∩ 𝑈 ) |
13 |
12
|
eqeq1i |
⊢ ( ( 𝑈 ∩ 𝑆 ) = { 0 } ↔ ( 𝑆 ∩ 𝑈 ) = { 0 } ) |
14 |
11 13
|
anbi12i |
⊢ ( ( ( 𝑇 ∩ 𝑆 ) = { 0 } ∧ ( 𝑈 ∩ 𝑆 ) = { 0 } ) ↔ ( ( 𝑆 ∩ 𝑇 ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) |
15 |
9 14
|
sylib |
⊢ ( 𝜑 → ( ( 𝑆 ∩ 𝑇 ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) |