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 )