Step |
Hyp |
Ref |
Expression |
1 |
|
trivnsgd.1 |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
trivnsgd.2 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
trivnsgd.3 |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
4 |
|
trivnsgd.4 |
⊢ ( 𝜑 → 𝐵 = { 0 } ) |
5 |
|
nsgsubg |
⊢ ( 𝑥 ∈ ( NrmSGrp ‘ 𝐺 ) → 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) |
6 |
5
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ ( NrmSGrp ‘ 𝐺 ) → 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) ) |
7 |
6
|
ssrdv |
⊢ ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) ⊆ ( SubGrp ‘ 𝐺 ) ) |
8 |
1 2 3 4
|
trivsubgsnd |
⊢ ( 𝜑 → ( SubGrp ‘ 𝐺 ) = { 𝐵 } ) |
9 |
7 8
|
sseqtrd |
⊢ ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) ⊆ { 𝐵 } ) |
10 |
1
|
nsgid |
⊢ ( 𝐺 ∈ Grp → 𝐵 ∈ ( NrmSGrp ‘ 𝐺 ) ) |
11 |
3 10
|
syl |
⊢ ( 𝜑 → 𝐵 ∈ ( NrmSGrp ‘ 𝐺 ) ) |
12 |
11
|
snssd |
⊢ ( 𝜑 → { 𝐵 } ⊆ ( NrmSGrp ‘ 𝐺 ) ) |
13 |
9 12
|
eqssd |
⊢ ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) = { 𝐵 } ) |