Metamath Proof Explorer


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 ( 𝑅𝑟 𝑆 → ( 𝑅 ∈ Field ↔ 𝑆 ∈ Field ) )

Proof

Step Hyp Ref Expression
1 ricdrng ( 𝑅𝑟 𝑆 → ( 𝑅 ∈ DivRing ↔ 𝑆 ∈ DivRing ) )
2 riccrng ( 𝑅𝑟 𝑆 → ( 𝑅 ∈ CRing ↔ 𝑆 ∈ CRing ) )
3 1 2 anbi12d ( 𝑅𝑟 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 ( 𝑅𝑟 𝑆 → ( 𝑅 ∈ Field ↔ 𝑆 ∈ Field ) )