Theorem oppr0

Description: Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014)

	Ref	Expression
Hypotheses	opprbas.1	$$⊢O={opp}_r\left(R\right)$$
	oppr0.2	$$⊢\underset{\dot{}}{0}=0_R$$
Assertion	oppr0	$$⊢\underset{\dot{}}{0}=0_O$$

Proof

Step	Hyp	Ref	Expression
1		opprbas.1	$$⊢O={opp}_r\left(R\right)$$
2		oppr0.2	$$⊢\underset{\dot{}}{0}=0_R$$
3		eqid	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_R$$
4		eqid	$$⊢{+}_R={+}_R$$
5	3 4 2	grpidval	$$⊢\underset{\dot{}}{0}=\left(\iota y|\left(y\in {\mathrm{Base}}_R\wedge \forall x\in {\mathrm{Base}}_R\left(y{+}_{R}x=x\wedge x{+}_{R}y=x\right)\right)\right)$$
6	1 3	opprbas	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_O$$
7	1 4	oppradd	$$⊢{+}_R={+}_O$$
8		eqid	$$⊢0_O=0_O$$
9	6 7 8	grpidval	$$⊢0_O=\left(\iota y|\left(y\in {\mathrm{Base}}_R\wedge \forall x\in {\mathrm{Base}}_R\left(y{+}_{R}x=x\wedge x{+}_{R}y=x\right)\right)\right)$$
10	5 9	eqtr4i	$$⊢\underset{\dot{}}{0}=0_O$$