Theorem oppgtset

Description: Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015)

	Ref	Expression
Hypotheses	oppgbas.1	$$⊢O={opp}_{𝑔}\left(R\right)$$
	oppgtset.2	$$⊢J=\mathrm{TopSet}\left(R\right)$$
Assertion	oppgtset	$$⊢J=\mathrm{TopSet}\left(O\right)$$

Proof

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