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 |
|
lsmdisj3b.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
7 |
|
lsmdisj3b.2 |
⊢ ( 𝜑 → 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) |
8 |
1 2 4 3 5
|
lsmdisj2b |
⊢ ( 𝜑 → ( ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ↔ ( ( 𝑆 ∩ ( 𝑈 ⊕ 𝑇 ) ) = { 0 } ∧ ( 𝑈 ∩ 𝑇 ) = { 0 } ) ) ) |
9 |
1 6
|
lsmcom2 |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) ) |
10 |
3 4 7 9
|
syl3anc |
⊢ ( 𝜑 → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) ) |
11 |
10
|
ineq2d |
⊢ ( 𝜑 → ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = ( 𝑆 ∩ ( 𝑈 ⊕ 𝑇 ) ) ) |
12 |
11
|
eqeq1d |
⊢ ( 𝜑 → ( ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } ↔ ( 𝑆 ∩ ( 𝑈 ⊕ 𝑇 ) ) = { 0 } ) ) |
13 |
|
incom |
⊢ ( 𝑇 ∩ 𝑈 ) = ( 𝑈 ∩ 𝑇 ) |
14 |
13
|
a1i |
⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) = ( 𝑈 ∩ 𝑇 ) ) |
15 |
14
|
eqeq1d |
⊢ ( 𝜑 → ( ( 𝑇 ∩ 𝑈 ) = { 0 } ↔ ( 𝑈 ∩ 𝑇 ) = { 0 } ) ) |
16 |
12 15
|
anbi12d |
⊢ ( 𝜑 → ( ( ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) ↔ ( ( 𝑆 ∩ ( 𝑈 ⊕ 𝑇 ) ) = { 0 } ∧ ( 𝑈 ∩ 𝑇 ) = { 0 } ) ) ) |
17 |
8 16
|
bitr4d |
⊢ ( 𝜑 → ( ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ↔ ( ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) ) ) |