Theorem riclcl

Description: Ring isomorphism implies the left side is a ring. (Contributed by AV, 23-Jul-2026)

	Ref	Expression
Assertion	riclcl	|- ( R ~=r S -> R e. Ring )

Proof

Step	Hyp	Ref	Expression
1		brric	|- ( R ~=r S <-> ( R RingIso S ) =/= (/) )
2		n0	|- ( ( R RingIso S ) =/= (/) <-> E. f f e. ( R RingIso S ) )
3	1 2	bitri	|- ( R ~=r S <-> E. f f e. ( R RingIso S ) )
4		rimrcl1	|- ( f e. ( R RingIso S ) -> R e. Ring )
5	4	exlimiv	|- ( E. f f e. ( R RingIso S ) -> R e. Ring )
6	3 5	sylbi	|- ( R ~=r S -> R e. Ring )