Step |
Hyp |
Ref |
Expression |
1 |
|
mndissubm.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
mndissubm.s |
⊢ 𝑆 = ( Base ‘ 𝐻 ) |
3 |
|
mndissubm.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
1 2 3
|
mndissubm |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) → ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) ) |
5 |
4
|
imp |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) |
6 |
|
simpl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) → 𝐺 ∈ Mnd ) |
7 |
|
3simpa |
⊢ ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) → ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) |
8 |
6 7
|
anim12i |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → ( 𝐺 ∈ Mnd ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ) |
9 |
8
|
biantrud |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ↔ ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ∧ ( 𝐺 ∈ Mnd ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ) ) ) |
10 |
|
an21 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ↔ ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ∧ ( 𝐺 ∈ Mnd ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ) ) |
11 |
9 10
|
bitr4di |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ↔ ( ( 𝐺 ∈ Mnd ∧ ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ) ) |
12 |
1 3
|
issubmndb |
⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ↔ ( ( 𝐺 ∈ Mnd ∧ ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ) |
13 |
11 12
|
bitr4di |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ↔ 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) ) |
14 |
5 13
|
mpbird |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) |
15 |
14
|
ex |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) → ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) → ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) ) |