Step |
Hyp |
Ref |
Expression |
1 |
|
drngpropd.1 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) ) |
2 |
|
drngpropd.2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) ) |
3 |
|
drngpropd.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ) |
4 |
|
drngpropd.4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ) |
5 |
1 2 3 4
|
drngpropd |
⊢ ( 𝜑 → ( 𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing ) ) |
6 |
1 2 3 4
|
crngpropd |
⊢ ( 𝜑 → ( 𝐾 ∈ CRing ↔ 𝐿 ∈ CRing ) ) |
7 |
5 6
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing ) ↔ ( 𝐿 ∈ DivRing ∧ 𝐿 ∈ CRing ) ) ) |
8 |
|
isfld |
⊢ ( 𝐾 ∈ Field ↔ ( 𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing ) ) |
9 |
|
isfld |
⊢ ( 𝐿 ∈ Field ↔ ( 𝐿 ∈ DivRing ∧ 𝐿 ∈ CRing ) ) |
10 |
7 8 9
|
3bitr4g |
⊢ ( 𝜑 → ( 𝐾 ∈ Field ↔ 𝐿 ∈ Field ) ) |