Metamath Proof Explorer


Theorem rimrhm

Description: A ring isomorphism is a homomorphism. Compare gimghm . (Contributed by AV, 22-Oct-2019) Remove hypotheses. (Revised by SN, 10-Jan-2025)

Ref Expression
Assertion rimrhm ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) )

Proof

Step Hyp Ref Expression
1 isrim0 ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ) )
2 1 simplbi ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) )