Theorem ldualneg

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

	Ref	Expression
Hypotheses	ldualneg.r	⊢ 𝑅 = ( Scalar ‘ 𝑊 )
	ldualneg.m	⊢ 𝑀 = ( invg ‘ 𝑅 )
	ldualneg.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
	ldualneg.s	⊢ 𝑆 = ( Scalar ‘ 𝐷 )
	ldualneg.n	⊢ 𝑁 = ( invg ‘ 𝑆 )
	ldualneg.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
Assertion	ldualneg	⊢ ( 𝜑 → 𝑁 = 𝑀 )

Proof

Step	Hyp	Ref	Expression
1		ldualneg.r	⊢ 𝑅 = ( Scalar ‘ 𝑊 )
2		ldualneg.m	⊢ 𝑀 = ( invg ‘ 𝑅 )
3		ldualneg.d	⊢ 𝐷 = ( LDual ‘ 𝑊 )
4		ldualneg.s	⊢ 𝑆 = ( Scalar ‘ 𝐷 )
5		ldualneg.n	⊢ 𝑁 = ( invg ‘ 𝑆 )
6		ldualneg.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 → ( invg ‘ 𝑆 ) = ( invg ‘ ( oppr ‘ 𝑅 ) ) )
10	7 2	opprneg	⊢ 𝑀 = ( invg ‘ ( oppr ‘ 𝑅 ) )
11	9 5 10	3eqtr4g	⊢ ( 𝜑 → 𝑁 = 𝑀 )