Step |
Hyp |
Ref |
Expression |
1 |
|
issubm2.b |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
2 |
|
issubm2.z |
⊢ 0 = ( 0g ‘ 𝑀 ) |
3 |
|
issubm2.h |
⊢ 𝐻 = ( 𝑀 ↾s 𝑆 ) |
4 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
5 |
1 2 4
|
issubm |
⊢ ( 𝑀 ∈ Mnd → ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝑆 ) ) ) |
6 |
1 4 2 3
|
issubmnd |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) → ( 𝐻 ∈ Mnd ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝑆 ) ) |
7 |
6
|
bicomd |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝑆 ↔ 𝐻 ∈ Mnd ) ) |
8 |
7
|
3expb |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝑆 ↔ 𝐻 ∈ Mnd ) ) |
9 |
8
|
pm5.32da |
⊢ ( 𝑀 ∈ Mnd → ( ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝑆 ) ↔ ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ) ) |
10 |
|
df-3an |
⊢ ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝑆 ) ↔ ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝑆 ) ) |
11 |
|
df-3an |
⊢ ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ 𝐻 ∈ Mnd ) ↔ ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ∧ 𝐻 ∈ Mnd ) ) |
12 |
9 10 11
|
3bitr4g |
⊢ ( 𝑀 ∈ Mnd → ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ 𝑆 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ 𝐻 ∈ Mnd ) ) ) |
13 |
5 12
|
bitrd |
⊢ ( 𝑀 ∈ Mnd → ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ 𝐻 ∈ Mnd ) ) ) |