Theorem opprneg

Description: The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014) (Revised by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	opprbas.1	⊢ 𝑂 = ( oppr ‘ 𝑅 )
	opprneg.2	⊢ 𝑁 = ( invg ‘ 𝑅 )
Assertion	opprneg	⊢ 𝑁 = ( invg ‘ 𝑂 )

Proof

Step	Hyp	Ref	Expression
1		opprbas.1	⊢ 𝑂 = ( oppr ‘ 𝑅 )
2		opprneg.2	⊢ 𝑁 = ( invg ‘ 𝑅 )
3		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
4		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
5		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
6	3 4 5 2	grpinvfval	⊢ 𝑁 = ( 𝑥 ∈ ( Base ‘ 𝑅 ) ↦ ( ℩ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑅 ) 𝑥 ) = ( 0g ‘ 𝑅 ) ) )
7	1 3	opprbas	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑂 )
8	1 4	oppradd	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑂 )
9	1 5	oppr0	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑂 )
10		eqid	⊢ ( invg ‘ 𝑂 ) = ( invg ‘ 𝑂 )
11	7 8 9 10	grpinvfval	⊢ ( invg ‘ 𝑂 ) = ( 𝑥 ∈ ( Base ‘ 𝑅 ) ↦ ( ℩ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑅 ) 𝑥 ) = ( 0g ‘ 𝑅 ) ) )
12	6 11	eqtr4i	⊢ 𝑁 = ( invg ‘ 𝑂 )