Theorem subrng0

Description: A subring always has the same additive identity. (Contributed by AV, 14-Feb-2025)

	Ref	Expression
Hypotheses	subrng0.1	$$⊢S=R{\upharpoonright }_{𝑠}A$$
	subrng0.2	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	subrng0	$$⊢A\in \mathrm{SubRng}\left(R\right)\to \underset{\dot{}}{0}=0_S$$

Proof

Step	Hyp	Ref	Expression
1		subrng0.1	$$⊢S=R{\upharpoonright }_{𝑠}A$$
2		subrng0.2	$$⊢\underset{\dot{}}{0}=0_R$$
3		subrngsubg	$$⊢A\in \mathrm{SubRng}\left(R\right)\to A\in \mathrm{SubGrp}\left(R\right)$$
4	1 2	subg0	$$⊢A\in \mathrm{SubGrp}\left(R\right)\to \underset{\dot{}}{0}=0_S$$
5	3 4	syl	$$⊢A\in \mathrm{SubRng}\left(R\right)\to \underset{\dot{}}{0}=0_S$$