Step |
Hyp |
Ref |
Expression |
1 |
|
issubmndb.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
issubmndb.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
eqid |
⊢ ( 𝐺 ↾s 𝑆 ) = ( 𝐺 ↾s 𝑆 ) |
4 |
1 2 3
|
issubm2 |
⊢ ( 𝐺 ∈ Mnd → ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) ) ) |
5 |
|
3anrot |
⊢ ( ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) ) |
6 |
|
3anass |
⊢ ( ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ↔ ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ) |
7 |
5 6
|
bitr3i |
⊢ ( ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) ↔ ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ) |
8 |
4 7
|
bitrdi |
⊢ ( 𝐺 ∈ Mnd → ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ↔ ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ) ) |
9 |
8
|
pm5.32i |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) ↔ ( 𝐺 ∈ Mnd ∧ ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ) ) |
10 |
|
submrcl |
⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → 𝐺 ∈ Mnd ) |
11 |
10
|
pm4.71ri |
⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ↔ ( 𝐺 ∈ Mnd ∧ 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) ) |
12 |
|
anass |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ↔ ( 𝐺 ∈ Mnd ∧ ( ( 𝐺 ↾s 𝑆 ) ∈ Mnd ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ) ) |
13 |
9 11 12
|
3bitr4i |
⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ↔ ( ( 𝐺 ∈ Mnd ∧ ( 𝐺 ↾s 𝑆 ) ∈ Mnd ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ) ) ) |