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	$$⊢O={opp}_r\left(R\right)$$
	opprbas.2	$$⊢B={\mathrm{Base}}_R$$
Assertion	opprbas	$$⊢B={\mathrm{Base}}_O$$

Proof

Step	Hyp	Ref	Expression
1		opprbas.1	$$⊢O={opp}_r\left(R\right)$$
2		opprbas.2	$$⊢B={\mathrm{Base}}_R$$
3		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
4		basendxnmulrndx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$
5	1 3 4	opprlem	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_O$$
6	2 5	eqtri	$$⊢B={\mathrm{Base}}_O$$