Step |
Hyp |
Ref |
Expression |
1 |
|
trivsubgsnd.1 |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
trivsubgsnd.2 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
trivsubgsnd.3 |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
4 |
|
trivsubgsnd.4 |
⊢ ( 𝜑 → 𝐵 = { 0 } ) |
5 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝐺 ∈ Grp ) |
6 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝐵 = { 0 } ) |
7 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) |
8 |
1 2 5 6 7
|
trivsubgd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑥 = 𝐵 ) |
9 |
|
velsn |
⊢ ( 𝑥 ∈ { 𝐵 } ↔ 𝑥 = 𝐵 ) |
10 |
8 9
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑥 ∈ { 𝐵 } ) |
11 |
10
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ ( SubGrp ‘ 𝐺 ) → 𝑥 ∈ { 𝐵 } ) ) |
12 |
11
|
ssrdv |
⊢ ( 𝜑 → ( SubGrp ‘ 𝐺 ) ⊆ { 𝐵 } ) |
13 |
1
|
subgid |
⊢ ( 𝐺 ∈ Grp → 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ) |
14 |
3 13
|
syl |
⊢ ( 𝜑 → 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ) |
15 |
14
|
snssd |
⊢ ( 𝜑 → { 𝐵 } ⊆ ( SubGrp ‘ 𝐺 ) ) |
16 |
12 15
|
eqssd |
⊢ ( 𝜑 → ( SubGrp ‘ 𝐺 ) = { 𝐵 } ) |