Theorem oppgtopn

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

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

Proof

Step	Hyp	Ref	Expression
1		oppgbas.1	$$⊢O={opp}_{𝑔}\left(R\right)$$
2		oppgtopn.2	$$⊢J=\mathrm{TopOpen}\left(R\right)$$
3		eqid	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_R$$
4		eqid	$$⊢\mathrm{TopSet}\left(R\right)=\mathrm{TopSet}\left(R\right)$$
5	3 4	topnval	$$⊢\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}{\mathrm{Base}}_R=\mathrm{TopOpen}\left(R\right)$$
6	1 3	oppgbas	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_O$$
7	1 4	oppgtset	$$⊢\mathrm{TopSet}\left(R\right)=\mathrm{TopSet}\left(O\right)$$
8	6 7	topnval	$$⊢\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}{\mathrm{Base}}_R=\mathrm{TopOpen}\left(O\right)$$
9	2 5 8	3eqtr2i	$$⊢J=\mathrm{TopOpen}\left(O\right)$$