Theorem ldual1

Description: The unit scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015)

	Ref	Expression
Hypotheses	ldual1.r	⊢ 𝑅 = ( Scalar ‘ 𝑊 )
	ldual1.u	⊢ 1 = ( 1r ‘ 𝑅 )
	ldual1.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
	ldual1.s	⊢ 𝑆 = ( Scalar ‘ 𝐷 )
	ldual1.i	⊢ 𝐼 = ( 1r ‘ 𝑆 )
	ldual1.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
Assertion	ldual1	⊢ ( 𝜑 → 𝐼 = 1 )

Proof

Step	Hyp	Ref	Expression
1		ldual1.r	⊢ 𝑅 = ( Scalar ‘ 𝑊 )
2		ldual1.u	⊢ 1 = ( 1r ‘ 𝑅 )
3		ldual1.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
4		ldual1.s	⊢ 𝑆 = ( Scalar ‘ 𝐷 )
5		ldual1.i	⊢ 𝐼 = ( 1r ‘ 𝑆 )
6		ldual1.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 → ( 1r ‘ 𝑆 ) = ( 1r ‘ ( oppr ‘ 𝑅 ) ) )
10	7 2	oppr1	⊢ 1 = ( 1r ‘ ( oppr ‘ 𝑅 ) )
11	9 5 10	3eqtr4g	⊢ ( 𝜑 → 𝐼 = 1 )