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 )