Theorem ldualsaddN

Description: Scalar addition for the dual of a vector space. (Contributed by NM, 24-Oct-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ldualsadd.f	|- F = ( Scalar ` W )
	ldualsadd.q	|- .+ = ( +g ` F )
	ldualsadd.d	|- D = ( LDual ` W )
	ldualsadd.r	|- R = ( Scalar ` D )
	ldualsadd.p	|- .+b = ( +g ` R )
	ldualsadd.w	|- ( ph -> W e. V )
Assertion	ldualsaddN	|- ( ph -> .+b = .+ )

Proof

Step	Hyp	Ref	Expression
1		ldualsadd.f	|- F = ( Scalar ` W )
2		ldualsadd.q	|- .+ = ( +g ` F )
3		ldualsadd.d	|- D = ( LDual ` W )
4		ldualsadd.r	|- R = ( Scalar ` D )
5		ldualsadd.p	|- .+b = ( +g ` R )
6		ldualsadd.w	|- ( ph -> W e. V )
7		eqid	|- ( oppR ` F ) = ( oppR ` F )
8	1 7 3 4 6	ldualsca	|- ( ph -> R = ( oppR ` F ) )
9	8	fveq2d	|- ( ph -> ( +g ` R ) = ( +g ` ( oppR ` F ) ) )
10	7 2	oppradd	|- .+ = ( +g ` ( oppR ` F ) )
11	9 5 10	3eqtr4g	|- ( ph -> .+b = .+ )