Theorem oppgbas

Description: Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015)

	Ref	Expression
Hypotheses	oppgbas.1	$$⊢O={opp}_{𝑔}\left(R\right)$$
	oppgbas.2	$$⊢B={\mathrm{Base}}_R$$
Assertion	oppgbas	$$⊢B={\mathrm{Base}}_O$$

Proof

Step	Hyp	Ref	Expression
1		oppgbas.1	$$⊢O={opp}_{𝑔}\left(R\right)$$
2		oppgbas.2	$$⊢B={\mathrm{Base}}_R$$
3		eqid	$$⊢{+}_R={+}_R$$
4	3 1	oppgval	$$⊢O=R\mathrm{sSet}⟨{+}_{\mathrm{ndx}},tpos{+}_R⟩$$
5		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
6		basendxnplusgndx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne {+}_{\mathrm{ndx}}$$
7	4 5 6	setsplusg	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_O$$
8	2 7	eqtri	$$⊢B={\mathrm{Base}}_O$$