Step |
Hyp |
Ref |
Expression |
1 |
|
mndissubm.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
mndissubm.s |
⊢ 𝑆 = ( Base ‘ 𝐻 ) |
3 |
|
mndissubm.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
|
simpr1 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → 𝑆 ⊆ 𝐵 ) |
5 |
|
simpr2 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → 0 ∈ 𝑆 ) |
6 |
|
mndmgm |
⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Mgm ) |
7 |
|
mndmgm |
⊢ ( 𝐻 ∈ Mnd → 𝐻 ∈ Mgm ) |
8 |
6 7
|
anim12i |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) → ( 𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm ) ) |
9 |
8
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) ∧ ( 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ) ) → ( 𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm ) ) |
10 |
|
3simpb |
⊢ ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) → ( 𝑆 ⊆ 𝐵 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) |
11 |
10
|
ad2antlr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) ∧ ( 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ) ) → ( 𝑆 ⊆ 𝐵 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) |
12 |
|
simpr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) ∧ ( 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ) ) → ( 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ) ) |
13 |
1 2
|
mgmsscl |
⊢ ( ( ( 𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm ) ∧ ( 𝑆 ⊆ 𝐵 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ∧ ( 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ) ) → ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ 𝑆 ) |
14 |
9 11 12 13
|
syl3anc |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) ∧ ( 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ) ) → ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ 𝑆 ) |
15 |
14
|
ralrimivva |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → ∀ 𝑎 ∈ 𝑆 ∀ 𝑏 ∈ 𝑆 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ 𝑆 ) |
16 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
17 |
1 3 16
|
issubm |
⊢ ( 𝐺 ∈ Mnd → ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀ 𝑎 ∈ 𝑆 ∀ 𝑏 ∈ 𝑆 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ 𝑆 ) ) ) |
18 |
17
|
ad2antrr |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀ 𝑎 ∈ 𝑆 ∀ 𝑏 ∈ 𝑆 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ 𝑆 ) ) ) |
19 |
4 5 15 18
|
mpbir3and |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) ) → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) |
20 |
19
|
ex |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd ) → ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( +g ‘ 𝐻 ) = ( ( +g ‘ 𝐺 ) ↾ ( 𝑆 × 𝑆 ) ) ) → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) ) |