Step |
Hyp |
Ref |
Expression |
1 |
|
grpissubg.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpissubg.s |
⊢ 𝑆 = ( Base ‘ 𝐻 ) |
3 |
1 2
|
grpissubg |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐻 ∈ Grp ) → ( ( 𝑆 ⊆ 𝐵 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) ) |
4 |
3
|
imp |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐻 ∈ Grp ) ∧ ( 𝑆 ⊆ 𝐵 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
5 |
|
ibar |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ) → ( ( 𝐺 ↾s 𝑆 ) ∈ Grp ↔ ( ( 𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ) ∧ ( 𝐺 ↾s 𝑆 ) ∈ Grp ) ) ) |
6 |
5
|
ad2ant2r |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐻 ∈ Grp ) ∧ ( 𝑆 ⊆ 𝐵 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → ( ( 𝐺 ↾s 𝑆 ) ∈ Grp ↔ ( ( 𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ) ∧ ( 𝐺 ↾s 𝑆 ) ∈ Grp ) ) ) |
7 |
|
df-3an |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ↾s 𝑆 ) ∈ Grp ) ↔ ( ( 𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ) ∧ ( 𝐺 ↾s 𝑆 ) ∈ Grp ) ) |
8 |
6 7
|
bitr4di |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐻 ∈ Grp ) ∧ ( 𝑆 ⊆ 𝐵 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → ( ( 𝐺 ↾s 𝑆 ) ∈ Grp ↔ ( 𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ↾s 𝑆 ) ∈ Grp ) ) ) |
9 |
1
|
issubg |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ ( 𝐺 ↾s 𝑆 ) ∈ Grp ) ) |
10 |
8 9
|
bitr4di |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐻 ∈ Grp ) ∧ ( 𝑆 ⊆ 𝐵 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → ( ( 𝐺 ↾s 𝑆 ) ∈ Grp ↔ 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) ) |
11 |
4 10
|
mpbird |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐻 ∈ Grp ) ∧ ( 𝑆 ⊆ 𝐵 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → ( 𝐺 ↾s 𝑆 ) ∈ Grp ) |
12 |
11
|
ex |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐻 ∈ Grp ) → ( ( 𝑆 ⊆ 𝐵 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) → ( 𝐺 ↾s 𝑆 ) ∈ Grp ) ) |