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	⊢ ✚ = ( +g ‘ 𝑅 )
	ldualsadd.w	⊢ ( 𝜑 → 𝑊 ∈ 𝑉 )
Assertion	ldualsaddN	⊢ ( 𝜑 → ✚ = + )

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	⊢ ✚ = ( +g ‘ 𝑅 )
6		ldualsadd.w	⊢ ( 𝜑 → 𝑊 ∈ 𝑉 )
7		eqid	⊢ ( oppr ‘ 𝐹 ) = ( oppr ‘ 𝐹 )
8	1 7 3 4 6	ldualsca	⊢ ( 𝜑 → 𝑅 = ( oppr ‘ 𝐹 ) )
9	8	fveq2d	⊢ ( 𝜑 → ( +g ‘ 𝑅 ) = ( +g ‘ ( oppr ‘ 𝐹 ) ) )
10	7 2	oppradd	⊢ + = ( +g ‘ ( oppr ‘ 𝐹 ) )
11	9 5 10	3eqtr4g	⊢ ( 𝜑 → ✚ = + )