Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 ) |
2 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
3 |
1 2
|
ismgmALT |
⊢ ( 𝑀 ∈ MgmALT → ( 𝑀 ∈ MgmALT ↔ ( +g ‘ 𝑀 ) clLaw ( Base ‘ 𝑀 ) ) ) |
4 |
|
fvex |
⊢ ( +g ‘ 𝑀 ) ∈ V |
5 |
|
fvex |
⊢ ( Base ‘ 𝑀 ) ∈ V |
6 |
|
iscllaw |
⊢ ( ( ( +g ‘ 𝑀 ) ∈ V ∧ ( Base ‘ 𝑀 ) ∈ V ) → ( ( +g ‘ 𝑀 ) clLaw ( Base ‘ 𝑀 ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) ) ) |
7 |
4 5 6
|
mp2an |
⊢ ( ( +g ‘ 𝑀 ) clLaw ( Base ‘ 𝑀 ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) ) |
8 |
1 2
|
ismgm |
⊢ ( 𝑀 ∈ MgmALT → ( 𝑀 ∈ Mgm ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) ) ) |
9 |
8
|
biimprd |
⊢ ( 𝑀 ∈ MgmALT → ( ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) → 𝑀 ∈ Mgm ) ) |
10 |
7 9
|
syl5bi |
⊢ ( 𝑀 ∈ MgmALT → ( ( +g ‘ 𝑀 ) clLaw ( Base ‘ 𝑀 ) → 𝑀 ∈ Mgm ) ) |
11 |
3 10
|
sylbid |
⊢ ( 𝑀 ∈ MgmALT → ( 𝑀 ∈ MgmALT → 𝑀 ∈ Mgm ) ) |
12 |
11
|
pm2.43i |
⊢ ( 𝑀 ∈ MgmALT → 𝑀 ∈ Mgm ) |
13 |
|
mgmplusgiopALT |
⊢ ( 𝑀 ∈ Mgm → ( +g ‘ 𝑀 ) clLaw ( Base ‘ 𝑀 ) ) |
14 |
1 2
|
ismgmALT |
⊢ ( 𝑀 ∈ Mgm → ( 𝑀 ∈ MgmALT ↔ ( +g ‘ 𝑀 ) clLaw ( Base ‘ 𝑀 ) ) ) |
15 |
13 14
|
mpbird |
⊢ ( 𝑀 ∈ Mgm → 𝑀 ∈ MgmALT ) |
16 |
12 15
|
impbii |
⊢ ( 𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm ) |