Theorem mgptopn

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

	Ref	Expression
Hypotheses	mgpbas.1	$$⊢M={\mathrm{mulGrp}}_R$$
	mgptopn.2	$$⊢J=\mathrm{TopOpen}\left(R\right)$$
Assertion	mgptopn	$$⊢J=\mathrm{TopOpen}\left(M\right)$$

Proof

Step	Hyp	Ref	Expression
1		mgpbas.1	$$⊢M={\mathrm{mulGrp}}_R$$
2		mgptopn.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	mgpbas	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_M$$
7	1	mgptset	$$⊢\mathrm{TopSet}\left(R\right)=\mathrm{TopSet}\left(M\right)$$
8	6 7	topnval	$$⊢\mathrm{TopSet}\left(R\right){\upharpoonright }_{𝑡}{\mathrm{Base}}_R=\mathrm{TopOpen}\left(M\right)$$
9	2 5 8	3eqtr2i	$$⊢J=\mathrm{TopOpen}\left(M\right)$$