Theorem mgptset

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

	Ref	Expression
Hypothesis	mgpbas.1	$$⊢M={\mathrm{mulGrp}}_R$$
Assertion	mgptset	$$⊢\mathrm{TopSet}\left(R\right)=\mathrm{TopSet}\left(M\right)$$

Proof

Step	Hyp	Ref	Expression
1		mgpbas.1	$$⊢M={\mathrm{mulGrp}}_R$$
2		eqid	$$⊢{\cdot }_R={\cdot }_R$$
3	1 2	mgpval	$$⊢M=R\mathrm{sSet}⟨{+}_{\mathrm{ndx}},{\cdot }_R⟩$$
4		tsetid	$$⊢\mathrm{TopSet}=\mathrm{Slot}\mathrm{TopSet}\left(\mathrm{ndx}\right)$$
5		tsetndxnplusgndx	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {+}_{\mathrm{ndx}}$$
6	3 4 5	setsplusg	$$⊢\mathrm{TopSet}\left(R\right)=\mathrm{TopSet}\left(M\right)$$