Step |
Hyp |
Ref |
Expression |
1 |
|
ablsimpnosubgd.1 |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
ablsimpnosubgd.2 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
ablsimpnosubgd.3 |
⊢ ( 𝜑 → 𝐺 ∈ Abel ) |
4 |
|
ablsimpnosubgd.4 |
⊢ ( 𝜑 → 𝐺 ∈ SimpGrp ) |
5 |
|
ablsimpnosubgd.5 |
⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
6 |
|
ablsimpnosubgd.6 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑆 ) |
7 |
|
ablsimpnosubgd.7 |
⊢ ( 𝜑 → ¬ 𝐴 = 0 ) |
8 |
|
elsni |
⊢ ( 𝐴 ∈ { 0 } → 𝐴 = 0 ) |
9 |
7 8
|
nsyl |
⊢ ( 𝜑 → ¬ 𝐴 ∈ { 0 } ) |
10 |
|
eleq2 |
⊢ ( 𝑆 = { 0 } → ( 𝐴 ∈ 𝑆 ↔ 𝐴 ∈ { 0 } ) ) |
11 |
6 10
|
syl5ibcom |
⊢ ( 𝜑 → ( 𝑆 = { 0 } → 𝐴 ∈ { 0 } ) ) |
12 |
9 11
|
mtod |
⊢ ( 𝜑 → ¬ 𝑆 = { 0 } ) |
13 |
12
|
pm2.21d |
⊢ ( 𝜑 → ( 𝑆 = { 0 } → 𝑆 = 𝐵 ) ) |
14 |
|
idd |
⊢ ( 𝜑 → ( 𝑆 = 𝐵 → 𝑆 = 𝐵 ) ) |
15 |
|
ablnsg |
⊢ ( 𝐺 ∈ Abel → ( NrmSGrp ‘ 𝐺 ) = ( SubGrp ‘ 𝐺 ) ) |
16 |
15
|
eqcomd |
⊢ ( 𝐺 ∈ Abel → ( SubGrp ‘ 𝐺 ) = ( NrmSGrp ‘ 𝐺 ) ) |
17 |
3 16
|
syl |
⊢ ( 𝜑 → ( SubGrp ‘ 𝐺 ) = ( NrmSGrp ‘ 𝐺 ) ) |
18 |
5 17
|
eleqtrd |
⊢ ( 𝜑 → 𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) ) |
19 |
1 2 4 18
|
simpgnsgeqd |
⊢ ( 𝜑 → ( 𝑆 = { 0 } ∨ 𝑆 = 𝐵 ) ) |
20 |
13 14 19
|
mpjaod |
⊢ ( 𝜑 → 𝑆 = 𝐵 ) |