Theorem subrngacl

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

	Ref	Expression
Hypothesis	subrngacl.p	$$⊢\underset{\dot{}}{+}={+}_R$$
Assertion	subrngacl	$$⊢\left(A\in \mathrm{SubRng}\left(R\right)\wedge X\in A\wedge Y\in A\right)\to X\underset{\dot{}}{+}Y\in A$$

Proof

Step	Hyp	Ref	Expression
1		subrngacl.p	$$⊢\underset{\dot{}}{+}={+}_R$$
2		subrngsubg	$$⊢A\in \mathrm{SubRng}\left(R\right)\to A\in \mathrm{SubGrp}\left(R\right)$$
3	1	subgcl	$$⊢\left(A\in \mathrm{SubGrp}\left(R\right)\wedge X\in A\wedge Y\in A\right)\to X\underset{\dot{}}{+}Y\in A$$
4	2 3	syl3an1	$$⊢\left(A\in \mathrm{SubRng}\left(R\right)\wedge X\in A\wedge Y\in A\right)\to X\underset{\dot{}}{+}Y\in A$$