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 ∈ ℕ
5		1ne2	⊢ 1 ≠ 2
6	1 3 4 5	mgplem	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑀 )
7	2 6	eqtri	⊢ 𝐵 = ( Base ‘ 𝑀 )