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		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
4	1 3	mgpval	⊢ 𝑀 = ( 𝑅 sSet ⟨ ( +g ‘ ndx ) , ( .r ‘ 𝑅 ) ⟩ )
5		baseid	⊢ Base = Slot ( Base ‘ ndx )
6		basendxnplusgndx	⊢ ( Base ‘ ndx ) ≠ ( +g ‘ ndx )
7	4 5 6	setsplusg	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑀 )
8	2 7	eqtri	⊢ 𝐵 = ( Base ‘ 𝑀 )