Theorem oppradd

Description: Addition operation 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 )
	oppradd.2	|- .+ = ( +g ` R )
Assertion	oppradd	|- .+ = ( +g ` O )

Proof

Step	Hyp	Ref	Expression
1		opprbas.1	|- O = ( oppR ` R )
2		oppradd.2	|- .+ = ( +g ` R )
3		plusgid	|- +g = Slot ( +g ` ndx )
4		plusgndxnmulrndx	|- ( +g ` ndx ) =/= ( .r ` ndx )
5	1 3 4	opprlem	|- ( +g ` R ) = ( +g ` O )
6	2 5	eqtri	|- .+ = ( +g ` O )