Theorem lcdsadd

Description: Scalar addition for the closed kernel vector space dual. (Contributed by NM, 6-Jun-2015)

	Ref	Expression
Hypotheses	lcdsadd.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcdsadd.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcdsadd.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
	lcdsadd.p	⊢ + = ( +g ‘ 𝐹 )
	lcdsadd.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	lcdsadd.s	⊢ 𝑆 = ( Scalar ‘ 𝐶 )
	lcdsadd.a	⊢ ✚ = ( +g ‘ 𝑆 )
	lcdsadd.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	lcdsadd	⊢ ( 𝜑 → ✚ = + )

Proof

Step	Hyp	Ref	Expression
1		lcdsadd.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcdsadd.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		lcdsadd.f	⊢ 𝐹 = ( Scalar ‘ 𝑈 )
4		lcdsadd.p	⊢ + = ( +g ‘ 𝐹 )
5		lcdsadd.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
6		lcdsadd.s	⊢ 𝑆 = ( Scalar ‘ 𝐶 )
7		lcdsadd.a	⊢ ✚ = ( +g ‘ 𝑆 )
8		lcdsadd.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		eqid	⊢ ( oppr ‘ 𝐹 ) = ( oppr ‘ 𝐹 )
10	1 2 3 9 5 6 8	lcdsca	⊢ ( 𝜑 → 𝑆 = ( oppr ‘ 𝐹 ) )
11	10	fveq2d	⊢ ( 𝜑 → ( +g ‘ 𝑆 ) = ( +g ‘ ( oppr ‘ 𝐹 ) ) )
12	9 4	oppradd	⊢ + = ( +g ‘ ( oppr ‘ 𝐹 ) )
13	11 7 12	3eqtr4g	⊢ ( 𝜑 → ✚ = + )