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 = ( mulGrp ` R )
	mgpbas.2	|- B = ( Base ` R )
Assertion	mgpbas	|- B = ( Base ` M )

Proof

Step	Hyp	Ref	Expression
1		mgpbas.1	|- M = ( mulGrp ` R )
2		mgpbas.2	|- B = ( Base ` R )
3		df-base	|- Base = Slot 1
4		1nn	|- 1 e. NN
5		1ne2	|- 1 =/= 2
6	1 3 4 5	mgplem	|- ( Base ` R ) = ( Base ` M )
7	2 6	eqtri	|- B = ( Base ` M )