Theorem subrg0

Description: A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014)

	Ref	Expression
Hypotheses	subrg0.1	$$⊢S=R{\upharpoonright }_{𝑠}A$$
	subrg0.2	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	subrg0	$$⊢A\in \mathrm{SubRing}\left(R\right)\to \underset{\dot{}}{0}=0_S$$

Proof

Step	Hyp	Ref	Expression
1		subrg0.1	$$⊢S=R{\upharpoonright }_{𝑠}A$$
2		subrg0.2	$$⊢\underset{\dot{}}{0}=0_R$$
3		subrgsubg	$$⊢A\in \mathrm{SubRing}\left(R\right)\to A\in \mathrm{SubGrp}\left(R\right)$$
4	1 2	subg0	$$⊢A\in \mathrm{SubGrp}\left(R\right)\to \underset{\dot{}}{0}=0_S$$
5	3 4	syl	$$⊢A\in \mathrm{SubRing}\left(R\right)\to \underset{\dot{}}{0}=0_S$$