Theorem opprbas

Description: Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014)

	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		df-base	⊢ Base = Slot 1
4		1nn	⊢ 1 ∈ ℕ
5		1lt3	⊢ 1 < 3
6	1 3 4 5	opprlem	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑂 )
7	2 6	eqtri	⊢ 𝐵 = ( Base ‘ 𝑂 )