Theorem oppgtset

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

	Ref	Expression
Hypotheses	oppgbas.1	⊢ 𝑂 = ( oppg ‘ 𝑅 )
	oppgtset.2	⊢ 𝐽 = ( TopSet ‘ 𝑅 )
Assertion	oppgtset	⊢ 𝐽 = ( TopSet ‘ 𝑂 )

Proof

Step	Hyp	Ref	Expression
1		oppgbas.1	⊢ 𝑂 = ( oppg ‘ 𝑅 )
2		oppgtset.2	⊢ 𝐽 = ( TopSet ‘ 𝑅 )
3		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
4	3 1	oppgval	⊢ 𝑂 = ( 𝑅 sSet ⟨ ( +g ‘ ndx ) , tpos ( +g ‘ 𝑅 ) ⟩ )
5		tsetid	⊢ TopSet = Slot ( TopSet ‘ ndx )
6		tsetndxnplusgndx	⊢ ( TopSet ‘ ndx ) ≠ ( +g ‘ ndx )
7	4 5 6	setsplusg	⊢ ( TopSet ‘ 𝑅 ) = ( TopSet ‘ 𝑂 )
8	2 7	eqtri	⊢ 𝐽 = ( TopSet ‘ 𝑂 )