Step |
Hyp |
Ref |
Expression |
1 |
|
ablpropd.1 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) ) |
2 |
|
ablpropd.2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) ) |
3 |
|
ablpropd.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ) |
4 |
1 2 3
|
mndpropd |
⊢ ( 𝜑 → ( 𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd ) ) |
5 |
3
|
oveqrspc2v |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ) ) → ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) ) |
6 |
3
|
oveqrspc2v |
⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵 ) ) → ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) |
7 |
6
|
ancom2s |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ) ) → ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) |
8 |
5 7
|
eqeq12d |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ) ) → ( ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ↔ ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) ) |
9 |
8
|
2ralbidva |
⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ↔ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) ) |
10 |
1
|
raleqdv |
⊢ ( 𝜑 → ( ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ↔ ∀ 𝑣 ∈ ( Base ‘ 𝐾 ) ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ) ) |
11 |
1 10
|
raleqbidv |
⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ↔ ∀ 𝑢 ∈ ( Base ‘ 𝐾 ) ∀ 𝑣 ∈ ( Base ‘ 𝐾 ) ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ) ) |
12 |
2
|
raleqdv |
⊢ ( 𝜑 → ( ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ↔ ∀ 𝑣 ∈ ( Base ‘ 𝐿 ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) ) |
13 |
2 12
|
raleqbidv |
⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ↔ ∀ 𝑢 ∈ ( Base ‘ 𝐿 ) ∀ 𝑣 ∈ ( Base ‘ 𝐿 ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) ) |
14 |
9 11 13
|
3bitr3d |
⊢ ( 𝜑 → ( ∀ 𝑢 ∈ ( Base ‘ 𝐾 ) ∀ 𝑣 ∈ ( Base ‘ 𝐾 ) ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ↔ ∀ 𝑢 ∈ ( Base ‘ 𝐿 ) ∀ 𝑣 ∈ ( Base ‘ 𝐿 ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) ) |
15 |
4 14
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝐾 ∈ Mnd ∧ ∀ 𝑢 ∈ ( Base ‘ 𝐾 ) ∀ 𝑣 ∈ ( Base ‘ 𝐾 ) ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ) ↔ ( 𝐿 ∈ Mnd ∧ ∀ 𝑢 ∈ ( Base ‘ 𝐿 ) ∀ 𝑣 ∈ ( Base ‘ 𝐿 ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) ) ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
17 |
|
eqid |
⊢ ( +g ‘ 𝐾 ) = ( +g ‘ 𝐾 ) |
18 |
16 17
|
iscmn |
⊢ ( 𝐾 ∈ CMnd ↔ ( 𝐾 ∈ Mnd ∧ ∀ 𝑢 ∈ ( Base ‘ 𝐾 ) ∀ 𝑣 ∈ ( Base ‘ 𝐾 ) ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ) ) |
19 |
|
eqid |
⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 ) |
20 |
|
eqid |
⊢ ( +g ‘ 𝐿 ) = ( +g ‘ 𝐿 ) |
21 |
19 20
|
iscmn |
⊢ ( 𝐿 ∈ CMnd ↔ ( 𝐿 ∈ Mnd ∧ ∀ 𝑢 ∈ ( Base ‘ 𝐿 ) ∀ 𝑣 ∈ ( Base ‘ 𝐿 ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) ) |
22 |
15 18 21
|
3bitr4g |
⊢ ( 𝜑 → ( 𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd ) ) |