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 |
|
lsmcntz.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
6 |
|
subgrcl |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
8 |
7
|
subgss |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 𝑈 ⊆ ( Base ‘ 𝐺 ) ) |
9 |
7 5
|
cntzsubg |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑈 ⊆ ( Base ‘ 𝐺 ) ) → ( 𝑍 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
10 |
6 8 9
|
syl2anc |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑍 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
11 |
4 10
|
syl |
⊢ ( 𝜑 → ( 𝑍 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
12 |
1
|
lsmlub |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑍 ‘ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑆 ⊆ ( 𝑍 ‘ 𝑈 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) ↔ ( 𝑆 ⊕ 𝑇 ) ⊆ ( 𝑍 ‘ 𝑈 ) ) ) |
13 |
2 3 11 12
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑆 ⊆ ( 𝑍 ‘ 𝑈 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) ↔ ( 𝑆 ⊕ 𝑇 ) ⊆ ( 𝑍 ‘ 𝑈 ) ) ) |
14 |
13
|
bicomd |
⊢ ( 𝜑 → ( ( 𝑆 ⊕ 𝑇 ) ⊆ ( 𝑍 ‘ 𝑈 ) ↔ ( 𝑆 ⊆ ( 𝑍 ‘ 𝑈 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) ) ) |