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 e. ( Base ` R ) |-> ( iota_ y e. ( 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 e. ( Base ` R ) |-> ( iota_ y e. ( Base ` R ) ( y ( +g ` R ) x ) = ( 0g ` R ) ) )
12	6 11	eqtr4i	|- N = ( invg ` O )