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