Theorem subrngbas

Description: Base set of a subring structure. (Contributed by AV, 14-Feb-2025)

	Ref	Expression
Hypothesis	subrng0.1	$$⊢S=R{\upharpoonright }_{𝑠}A$$
Assertion	subrngbas	$$⊢A\in \mathrm{SubRng}\left(R\right)\to A={\mathrm{Base}}_S$$

Proof

Step	Hyp	Ref	Expression
1		subrng0.1	$$⊢S=R{\upharpoonright }_{𝑠}A$$
2		subrngsubg	$$⊢A\in \mathrm{SubRng}\left(R\right)\to A\in \mathrm{SubGrp}\left(R\right)$$
3	1	subgbas	$$⊢A\in \mathrm{SubGrp}\left(R\right)\to A={\mathrm{Base}}_S$$
4	2 3	syl	$$⊢A\in \mathrm{SubRng}\left(R\right)\to A={\mathrm{Base}}_S$$