Theorem mgpbas

Description: Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014) (Revised by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Hypotheses	mgpbas.1	$$⊢M={\mathrm{mulGrp}}_R$$
	mgpbas.2	$$⊢B={\mathrm{Base}}_R$$
Assertion	mgpbas	$$⊢B={\mathrm{Base}}_M$$

Proof

Step	Hyp	Ref	Expression
1		mgpbas.1	$$⊢M={\mathrm{mulGrp}}_R$$
2		mgpbas.2	$$⊢B={\mathrm{Base}}_R$$
3		eqid	$$⊢{\cdot }_R={\cdot }_R$$
4	1 3	mgpval	$$⊢M=R\mathrm{sSet}⟨{+}_{\mathrm{ndx}},{\cdot }_R⟩$$
5		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
6		basendxnplusgndx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne {+}_{\mathrm{ndx}}$$
7	4 5 6	setsplusg	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_M$$
8	2 7	eqtri	$$⊢B={\mathrm{Base}}_M$$