Theorem mgptopn

Description: Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Hypotheses	mgpbas.1	⊢ 𝑀 = ( mulGrp ‘ 𝑅 )
	mgptopn.2	⊢ 𝐽 = ( TopOpen ‘ 𝑅 )
Assertion	mgptopn	⊢ 𝐽 = ( TopOpen ‘ 𝑀 )

Proof

Step	Hyp	Ref	Expression
1		mgpbas.1	⊢ 𝑀 = ( mulGrp ‘ 𝑅 )
2		mgptopn.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	mgpbas	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑀 )
7	1	mgptset	⊢ ( TopSet ‘ 𝑅 ) = ( TopSet ‘ 𝑀 )
8	6 7	topnval	⊢ ( ( TopSet ‘ 𝑅 ) ↾t ( Base ‘ 𝑅 ) ) = ( TopOpen ‘ 𝑀 )
9	2 5 8	3eqtr2i	⊢ 𝐽 = ( TopOpen ‘ 𝑀 )