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 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
7 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) |
8 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
9 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ) |
10 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( 𝑆 ∩ 𝑇 ) = { 0 } ) |
11 |
1 6 7 8 5 9 10
|
lsmdisj2 |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ) |
12 |
1 6 7 8 5 9
|
lsmdisj |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( ( 𝑆 ∩ 𝑈 ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) ) |
13 |
12
|
simpld |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( 𝑆 ∩ 𝑈 ) = { 0 } ) |
14 |
11 13
|
jca |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) |
15 |
|
incom |
⊢ ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = ( 𝑈 ∩ ( 𝑆 ⊕ 𝑇 ) ) |
16 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
17 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
18 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) |
19 |
|
incom |
⊢ ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 ) = ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) |
20 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ) |
21 |
19 20
|
syl5eq |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 ) = { 0 } ) |
22 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( 𝑆 ∩ 𝑈 ) = { 0 } ) |
23 |
1 16 17 18 5 21 22
|
lsmdisj2 |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( 𝑈 ∩ ( 𝑆 ⊕ 𝑇 ) ) = { 0 } ) |
24 |
15 23
|
syl5eq |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ) |
25 |
|
incom |
⊢ ( 𝑆 ∩ 𝑇 ) = ( 𝑇 ∩ 𝑆 ) |
26 |
1 18 16 17 5 20
|
lsmdisjr |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( ( 𝑇 ∩ 𝑆 ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) ) |
27 |
26
|
simpld |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( 𝑇 ∩ 𝑆 ) = { 0 } ) |
28 |
25 27
|
syl5eq |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( 𝑆 ∩ 𝑇 ) = { 0 } ) |
29 |
24 28
|
jca |
⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) |
30 |
14 29
|
impbida |
⊢ ( 𝜑 → ( ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ↔ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) ) |