Theorem rimrhm

Description: A ring isomorphism is a homomorphism. Compare gimghm . (Contributed by Jeff Madsen, 16-Jun-2011) (Revised 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 ) )