Theorem rngacl

Description: Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025)

	Ref	Expression
Hypotheses	rngacl.b	|- B = ( Base ` R )
	rngacl.p	|- .+ = ( +g ` R )
Assertion	rngacl	|- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )

Proof

Step	Hyp	Ref	Expression
1		rngacl.b	|- B = ( Base ` R )
2		rngacl.p	|- .+ = ( +g ` R )
3		rnggrp	|- ( R e. Rng -> R e. Grp )
4	1 2	grpcl	|- ( ( R e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
5	3 4	syl3an1	|- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )