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 )