Theorem opprbas

Description: Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014) (Proof shortened by AV, 6-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		opprbas.1	⊢ 𝑂 = ( oppr ‘ 𝑅 )
2		opprbas.2	⊢ 𝐵 = ( Base ‘ 𝑅 )
3		baseid	⊢ Base = Slot ( Base ‘ ndx )
4		basendxnmulrndx	⊢ ( Base ‘ ndx ) ≠ ( .r ‘ ndx )
5	1 3 4	opprlem	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑂 )
6	2 5	eqtri	⊢ 𝐵 = ( Base ‘ 𝑂 )