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 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		rngacl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		rngacl.p	⊢ + = ( +g ‘ 𝑅 )
3		rnggrp	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Grp )
4	1 2	grpcl	⊢ ( ( 𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
5	3 4	syl3an1	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )