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		df-base	$$⊢\mathrm{Base}=\mathrm{Slot}1$$
4		1nn	$$⊢1\in \mathbb{N}$$
5		1ne2	$$⊢1\ne 2$$
6	1 3 4 5	oppglem	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_O$$
7	2 6	eqtri	$$⊢B={\mathrm{Base}}_O$$