Theorem ldual0vcl

Description: The dual zero vector is a functional. (Contributed by NM, 5-Mar-2015)

	Ref	Expression
Hypotheses	ldualv0cl.f	⊢ 𝐹 = ( LFnl ‘ 𝑊 )
	ldualv0cl.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
	ldualv0cl.o	⊢ 0 = ( 0g ‘ 𝐷 )
	ldualv0cl.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
Assertion	ldual0vcl	⊢ ( 𝜑 → 0 ∈ 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		ldualv0cl.f	⊢ 𝐹 = ( LFnl ‘ 𝑊 )
2		ldualv0cl.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
3		ldualv0cl.o	⊢ 0 = ( 0g ‘ 𝐷 )
4		ldualv0cl.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
5		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
6		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
7		eqid	⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
8	5 6 7 2 3 4	ldual0v	⊢ ( 𝜑 → 0 = ( ( Base ‘ 𝑊 ) × { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) )
9	6 7 5 1	lfl0f	⊢ ( 𝑊 ∈ LMod → ( ( Base ‘ 𝑊 ) × { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ∈ 𝐹 )
10	4 9	syl	⊢ ( 𝜑 → ( ( Base ‘ 𝑊 ) × { ( 0g ‘ ( Scalar ‘ 𝑊 ) ) } ) ∈ 𝐹 )
11	8 10	eqeltrd	⊢ ( 𝜑 → 0 ∈ 𝐹 )