Theorem subrgmcl

Description: A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014) (Proof shortened by AV, 30-Mar-2025)

	Ref	Expression
Hypothesis	subrgmcl.p	⊢ · = ( .r ‘ 𝑅 )
Assertion	subrgmcl	⊢ ( ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 · 𝑌 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		subrgmcl.p	⊢ · = ( .r ‘ 𝑅 )
2		subrgsubrng	⊢ ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) → 𝐴 ∈ ( SubRng ‘ 𝑅 ) )
3	1	subrngmcl	⊢ ( ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 · 𝑌 ) ∈ 𝐴 )
4	2 3	syl3an1	⊢ ( ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 · 𝑌 ) ∈ 𝐴 )