Theorem oppgbas

Description: Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015)

	Ref	Expression
Hypotheses	oppgbas.1	⊢ 𝑂 = ( oppg ‘ 𝑅 )
	oppgbas.2	⊢ 𝐵 = ( Base ‘ 𝑅 )
Assertion	oppgbas	⊢ 𝐵 = ( Base ‘ 𝑂 )

Proof

Step	Hyp	Ref	Expression
1		oppgbas.1	⊢ 𝑂 = ( oppg ‘ 𝑅 )
2		oppgbas.2	⊢ 𝐵 = ( Base ‘ 𝑅 )
3		df-base	⊢ Base = Slot 1
4		1nn	⊢ 1 ∈ ℕ
5		1ne2	⊢ 1 ≠ 2
6	1 3 4 5	oppglem	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑂 )
7	2 6	eqtri	⊢ 𝐵 = ( Base ‘ 𝑂 )