Theorem ldual0

Description: The zero scalar of the dual of a vector space. (Contributed by NM, 28-Dec-2014)

	Ref	Expression
Hypotheses	ldual0.r	⊢ 𝑅 = ( Scalar ‘ 𝑊 )
	ldual0.z	⊢ 0 = ( 0g ‘ 𝑅 )
	ldual0.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
	ldual0.s	⊢ 𝑆 = ( Scalar ‘ 𝐷 )
	ldual0.o	⊢ 𝑂 = ( 0g ‘ 𝑆 )
	ldual0.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
Assertion	ldual0	⊢ ( 𝜑 → 𝑂 = 0 )

Proof

Step	Hyp	Ref	Expression
1		ldual0.r	⊢ 𝑅 = ( Scalar ‘ 𝑊 )
2		ldual0.z	⊢ 0 = ( 0g ‘ 𝑅 )
3		ldual0.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
4		ldual0.s	⊢ 𝑆 = ( Scalar ‘ 𝐷 )
5		ldual0.o	⊢ 𝑂 = ( 0g ‘ 𝑆 )
6		ldual0.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
7		eqid	⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 )
8	1 7 3 4 6	ldualsca	⊢ ( 𝜑 → 𝑆 = ( oppr ‘ 𝑅 ) )
9	8	fveq2d	⊢ ( 𝜑 → ( 0g ‘ 𝑆 ) = ( 0g ‘ ( oppr ‘ 𝑅 ) ) )
10	7 2	oppr0	⊢ 0 = ( 0g ‘ ( oppr ‘ 𝑅 ) )
11	9 5 10	3eqtr4g	⊢ ( 𝜑 → 𝑂 = 0 )