Theorem subrngrng

Description: A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025)

	Ref	Expression
Hypothesis	subrngrng.1	|- S = ( R |`s A )
Assertion	subrngrng	|- ( A e. ( SubRng ` R ) -> S e. Rng )

Proof

Step	Hyp	Ref	Expression
1		subrngrng.1	|- S = ( R |`s A )
2		simp2	|- ( ( R e. Rng /\ ( R |`s A ) e. Rng /\ A C_ ( Base ` R ) ) -> ( R |`s A ) e. Rng )
3		eqid	|- ( Base ` R ) = ( Base ` R )
4	3	issubrng	|- ( A e. ( SubRng ` R ) <-> ( R e. Rng /\ ( R |`s A ) e. Rng /\ A C_ ( Base ` R ) ) )
5	1	eleq1i	|- ( S e. Rng <-> ( R |`s A ) e. Rng )
6	2 4 5	3imtr4i	|- ( A e. ( SubRng ` R ) -> S e. Rng )