Theorem nvmfval

Description: Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 11-Sep-2007) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvmval.1	|- X = ( BaseSet ` U )
	nvmval.2	|- G = ( +v ` U )
	nvmval.4	|- S = ( .sOLD ` U )
	nvmval.3	|- M = ( -v ` U )
Assertion	nvmfval	|- ( U e. NrmCVec -> M = ( x e. X , y e. X |-> ( x G ( -u 1 S y ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		nvmval.1	|- X = ( BaseSet ` U )
2		nvmval.2	|- G = ( +v ` U )
3		nvmval.4	|- S = ( .sOLD ` U )
4		nvmval.3	|- M = ( -v ` U )
5	2	nvgrp	|- ( U e. NrmCVec -> G e. GrpOp )
6	1 2	bafval	|- X = ran G
7		eqid	|- ( inv ` G ) = ( inv ` G )
8	2 4	vsfval	|- M = ( /g ` G )
9	6 7 8	grpodivfval	|- ( G e. GrpOp -> M = ( x e. X , y e. X |-> ( x G ( ( inv ` G ) ` y ) ) ) )
10	5 9	syl	|- ( U e. NrmCVec -> M = ( x e. X , y e. X |-> ( x G ( ( inv ` G ) ` y ) ) ) )
11	1 2 3 7	nvinv	|- ( ( U e. NrmCVec /\ y e. X ) -> ( -u 1 S y ) = ( ( inv ` G ) ` y ) )
12	11	3adant2	|- ( ( U e. NrmCVec /\ x e. X /\ y e. X ) -> ( -u 1 S y ) = ( ( inv ` G ) ` y ) )
13	12	oveq2d	|- ( ( U e. NrmCVec /\ x e. X /\ y e. X ) -> ( x G ( -u 1 S y ) ) = ( x G ( ( inv ` G ) ` y ) ) )
14	13	mpoeq3dva	|- ( U e. NrmCVec -> ( x e. X , y e. X |-> ( x G ( -u 1 S y ) ) ) = ( x e. X , y e. X |-> ( x G ( ( inv ` G ) ` y ) ) ) )
15	10 14	eqtr4d	|- ( U e. NrmCVec -> M = ( x e. X , y e. X |-> ( x G ( -u 1 S y ) ) ) )