Theorem ldualsbase

Description: Base set of scalar ring for the dual of a vector space. (Contributed by NM, 24-Oct-2014)

	Ref	Expression
Hypotheses	ldualsbase.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	ldualsbase.l	⊢ 𝐿 = ( Base ‘ 𝐹 )
	ldualsbase.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
	ldualsbase.r	⊢ 𝑅 = ( Scalar ‘ 𝐷 )
	ldualsbase.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	ldualsbase.w	⊢ ( 𝜑 → 𝑊 ∈ 𝑉 )
Assertion	ldualsbase	⊢ ( 𝜑 → 𝐾 = 𝐿 )

Proof

Step	Hyp	Ref	Expression
1		ldualsbase.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2		ldualsbase.l	⊢ 𝐿 = ( Base ‘ 𝐹 )
3		ldualsbase.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
4		ldualsbase.r	⊢ 𝑅 = ( Scalar ‘ 𝐷 )
5		ldualsbase.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
6		ldualsbase.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 → ( Base ‘ 𝑅 ) = ( Base ‘ ( oppr ‘ 𝐹 ) ) )
10	7 2	opprbas	⊢ 𝐿 = ( Base ‘ ( oppr ‘ 𝐹 ) )
11	9 5 10	3eqtr4g	⊢ ( 𝜑 → 𝐾 = 𝐿 )