Description: A ring is a field if and only if an isomorphic ring is a field. (Contributed by SN, 18-Feb-2025)
Ref | Expression | ||
---|---|---|---|
Assertion | ricfld | ⊢ ( 𝑅 ≃𝑟 𝑆 → ( 𝑅 ∈ Field ↔ 𝑆 ∈ Field ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ricdrng | ⊢ ( 𝑅 ≃𝑟 𝑆 → ( 𝑅 ∈ DivRing ↔ 𝑆 ∈ DivRing ) ) | |
2 | riccrng | ⊢ ( 𝑅 ≃𝑟 𝑆 → ( 𝑅 ∈ CRing ↔ 𝑆 ∈ CRing ) ) | |
3 | 1 2 | anbi12d | ⊢ ( 𝑅 ≃𝑟 𝑆 → ( ( 𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing ) ↔ ( 𝑆 ∈ DivRing ∧ 𝑆 ∈ CRing ) ) ) |
4 | isfld | ⊢ ( 𝑅 ∈ Field ↔ ( 𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing ) ) | |
5 | isfld | ⊢ ( 𝑆 ∈ Field ↔ ( 𝑆 ∈ DivRing ∧ 𝑆 ∈ CRing ) ) | |
6 | 3 4 5 | 3bitr4g | ⊢ ( 𝑅 ≃𝑟 𝑆 → ( 𝑅 ∈ Field ↔ 𝑆 ∈ Field ) ) |