Step |
Hyp |
Ref |
Expression |
1 |
|
lsmsubg.p |
⊢ ⊕ = ( LSSum ‘ 𝐺 ) |
2 |
|
lsmsubg.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
3 |
|
simp3 |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) |
4 |
3
|
sselda |
⊢ ( ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) ∧ 𝑎 ∈ 𝑇 ) → 𝑎 ∈ ( 𝑍 ‘ 𝑈 ) ) |
5 |
4
|
adantrr |
⊢ ( ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) ∧ ( 𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈 ) ) → 𝑎 ∈ ( 𝑍 ‘ 𝑈 ) ) |
6 |
|
simprr |
⊢ ( ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) ∧ ( 𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈 ) ) → 𝑏 ∈ 𝑈 ) |
7 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
8 |
7 2
|
cntzi |
⊢ ( ( 𝑎 ∈ ( 𝑍 ‘ 𝑈 ) ∧ 𝑏 ∈ 𝑈 ) → ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑏 ( +g ‘ 𝐺 ) 𝑎 ) ) |
9 |
5 6 8
|
syl2anc |
⊢ ( ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) ∧ ( 𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈 ) ) → ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑏 ( +g ‘ 𝐺 ) 𝑎 ) ) |
10 |
9
|
eqeq2d |
⊢ ( ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) ∧ ( 𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑈 ) ) → ( 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ↔ 𝑥 = ( 𝑏 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
11 |
10
|
2rexbidva |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → ( ∃ 𝑎 ∈ 𝑇 ∃ 𝑏 ∈ 𝑈 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ↔ ∃ 𝑎 ∈ 𝑇 ∃ 𝑏 ∈ 𝑈 𝑥 = ( 𝑏 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
12 |
|
rexcom |
⊢ ( ∃ 𝑎 ∈ 𝑇 ∃ 𝑏 ∈ 𝑈 𝑥 = ( 𝑏 ( +g ‘ 𝐺 ) 𝑎 ) ↔ ∃ 𝑏 ∈ 𝑈 ∃ 𝑎 ∈ 𝑇 𝑥 = ( 𝑏 ( +g ‘ 𝐺 ) 𝑎 ) ) |
13 |
11 12
|
bitrdi |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → ( ∃ 𝑎 ∈ 𝑇 ∃ 𝑏 ∈ 𝑈 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ↔ ∃ 𝑏 ∈ 𝑈 ∃ 𝑎 ∈ 𝑇 𝑥 = ( 𝑏 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
14 |
7 1
|
lsmelval |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑎 ∈ 𝑇 ∃ 𝑏 ∈ 𝑈 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ) ) |
15 |
14
|
3adant3 |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑎 ∈ 𝑇 ∃ 𝑏 ∈ 𝑈 𝑥 = ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ) ) |
16 |
7 1
|
lsmelval |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑥 ∈ ( 𝑈 ⊕ 𝑇 ) ↔ ∃ 𝑏 ∈ 𝑈 ∃ 𝑎 ∈ 𝑇 𝑥 = ( 𝑏 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
17 |
16
|
ancoms |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑥 ∈ ( 𝑈 ⊕ 𝑇 ) ↔ ∃ 𝑏 ∈ 𝑈 ∃ 𝑎 ∈ 𝑇 𝑥 = ( 𝑏 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
18 |
17
|
3adant3 |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → ( 𝑥 ∈ ( 𝑈 ⊕ 𝑇 ) ↔ ∃ 𝑏 ∈ 𝑈 ∃ 𝑎 ∈ 𝑇 𝑥 = ( 𝑏 ( +g ‘ 𝐺 ) 𝑎 ) ) ) |
19 |
13 15 18
|
3bitr4d |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ 𝑥 ∈ ( 𝑈 ⊕ 𝑇 ) ) ) |
20 |
19
|
eqrdv |
⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) ) |