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	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
Assertion	subrgmcl	$$⊢\left(A\in \mathrm{SubRing}\left(R\right)\wedge X\in A\wedge Y\in A\right)\to X\underset{\dot{}}{·}Y\in A$$

Proof

Step	Hyp	Ref	Expression
1		subrgmcl.p	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
2		subrgsubrng	Could not format ( A e. ( SubRing ` R ) -> A e. ( SubRng ` R ) ) : No typesetting found for |- ( A e. ( SubRing ` R ) -> A e. ( SubRng ` R ) ) with typecode |-
3	1	subrngmcl	Could not format ( ( A e. ( SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A ) : No typesetting found for |- ( ( A e. ( SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A ) with typecode |-
4	2 3	syl3an1	$$⊢\left(A\in \mathrm{SubRing}\left(R\right)\wedge X\in A\wedge Y\in A\right)\to X\underset{\dot{}}{·}Y\in A$$