Theorem subrngacl

Description: A subring is closed under addition. (Contributed by AV, 14-Feb-2025)

	Ref	Expression
Hypothesis	subrngacl.p	⊢ + = ( +g ‘ 𝑅 )
Assertion	subrngacl	⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 + 𝑌 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		subrngacl.p	⊢ + = ( +g ‘ 𝑅 )
2		subrngsubg	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → 𝐴 ∈ ( SubGrp ‘ 𝑅 ) )
3	1	subgcl	⊢ ( ( 𝐴 ∈ ( SubGrp ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 + 𝑌 ) ∈ 𝐴 )
4	2 3	syl3an1	⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 + 𝑌 ) ∈ 𝐴 )