Theorem oppgtopn

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

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

Proof

Step	Hyp	Ref	Expression
1		oppgbas.1	⊢ 𝑂 = ( oppg ‘ 𝑅 )
2		oppgtopn.2	⊢ 𝐽 = ( TopOpen ‘ 𝑅 )
3		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
4		eqid	⊢ ( TopSet ‘ 𝑅 ) = ( TopSet ‘ 𝑅 )
5	3 4	topnval	⊢ ( ( TopSet ‘ 𝑅 ) ↾t ( Base ‘ 𝑅 ) ) = ( TopOpen ‘ 𝑅 )
6	1 3	oppgbas	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑂 )
7	1 4	oppgtset	⊢ ( TopSet ‘ 𝑅 ) = ( TopSet ‘ 𝑂 )
8	6 7	topnval	⊢ ( ( TopSet ‘ 𝑅 ) ↾t ( Base ‘ 𝑅 ) ) = ( TopOpen ‘ 𝑂 )
9	2 5 8	3eqtr2i	⊢ 𝐽 = ( TopOpen ‘ 𝑂 )