Step |
Hyp |
Ref |
Expression |
1 |
|
lsmub1.p |
⊢ ⊕ = ( LSSum ‘ 𝐺 ) |
2 |
|
simp3 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
3 |
1
|
lsmless12 |
⊢ ( ( ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) ∧ ( 𝑆 ⊆ 𝑈 ∧ 𝑇 ⊆ 𝑈 ) ) → ( 𝑆 ⊕ 𝑇 ) ⊆ ( 𝑈 ⊕ 𝑈 ) ) |
4 |
3
|
ex |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑆 ⊆ 𝑈 ∧ 𝑇 ⊆ 𝑈 ) → ( 𝑆 ⊕ 𝑇 ) ⊆ ( 𝑈 ⊕ 𝑈 ) ) ) |
5 |
2 2 4
|
syl2anc |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑆 ⊆ 𝑈 ∧ 𝑇 ⊆ 𝑈 ) → ( 𝑆 ⊕ 𝑇 ) ⊆ ( 𝑈 ⊕ 𝑈 ) ) ) |
6 |
1
|
lsmidm |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑈 ⊕ 𝑈 ) = 𝑈 ) |
7 |
6
|
3ad2ant3 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑈 ⊕ 𝑈 ) = 𝑈 ) |
8 |
7
|
sseq2d |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑆 ⊕ 𝑇 ) ⊆ ( 𝑈 ⊕ 𝑈 ) ↔ ( 𝑆 ⊕ 𝑇 ) ⊆ 𝑈 ) ) |
9 |
5 8
|
sylibd |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑆 ⊆ 𝑈 ∧ 𝑇 ⊆ 𝑈 ) → ( 𝑆 ⊕ 𝑇 ) ⊆ 𝑈 ) ) |
10 |
1
|
lsmub1 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑆 ⊆ ( 𝑆 ⊕ 𝑇 ) ) |
11 |
10
|
3adant3 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑆 ⊆ ( 𝑆 ⊕ 𝑇 ) ) |
12 |
|
sstr2 |
⊢ ( 𝑆 ⊆ ( 𝑆 ⊕ 𝑇 ) → ( ( 𝑆 ⊕ 𝑇 ) ⊆ 𝑈 → 𝑆 ⊆ 𝑈 ) ) |
13 |
11 12
|
syl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑆 ⊕ 𝑇 ) ⊆ 𝑈 → 𝑆 ⊆ 𝑈 ) ) |
14 |
1
|
lsmub2 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑇 ⊆ ( 𝑆 ⊕ 𝑇 ) ) |
15 |
14
|
3adant3 |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑇 ⊆ ( 𝑆 ⊕ 𝑇 ) ) |
16 |
|
sstr2 |
⊢ ( 𝑇 ⊆ ( 𝑆 ⊕ 𝑇 ) → ( ( 𝑆 ⊕ 𝑇 ) ⊆ 𝑈 → 𝑇 ⊆ 𝑈 ) ) |
17 |
15 16
|
syl |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑆 ⊕ 𝑇 ) ⊆ 𝑈 → 𝑇 ⊆ 𝑈 ) ) |
18 |
13 17
|
jcad |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑆 ⊕ 𝑇 ) ⊆ 𝑈 → ( 𝑆 ⊆ 𝑈 ∧ 𝑇 ⊆ 𝑈 ) ) ) |
19 |
9 18
|
impbid |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑆 ⊆ 𝑈 ∧ 𝑇 ⊆ 𝑈 ) ↔ ( 𝑆 ⊕ 𝑇 ) ⊆ 𝑈 ) ) |