Theorem rimrcl1

Description: Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025)

	Ref	Expression
Assertion	rimrcl1	⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝑅 ∈ Ring )

Proof

Step	Hyp	Ref	Expression
1		rimrhm	⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) )
2		rhmrcl1	⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝑅 ∈ Ring )
3	1 2	syl	⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝑅 ∈ Ring )