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		df-base	$$⊢\mathrm{Base}=\mathrm{Slot}1$$
4		1nn	$$⊢1\in \mathbb{N}$$
5		1ne2	$$⊢1\ne 2$$
6	1 3 4 5	mgplem	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_M$$
7	2 6	eqtri	$$⊢B={\mathrm{Base}}_M$$