Theorem ricfld

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 ) )

Proof

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 ) )