Theorem lcdneg

Description: The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 11-Jun-2015)

	Ref	Expression
Hypotheses	lcdneg.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcdneg.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcdneg.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	lcdneg.m	⊢ 𝑀 = ( invg ‘ 𝑅 )
	lcdneg.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	lcdneg.s	⊢ 𝑆 = ( Scalar ‘ 𝐶 )
	lcdneg.n	⊢ 𝑁 = ( invg ‘ 𝑆 )
	lcdneg.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	lcdneg	⊢ ( 𝜑 → 𝑁 = 𝑀 )

Proof

Step	Hyp	Ref	Expression
1		lcdneg.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcdneg.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		lcdneg.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
4		lcdneg.m	⊢ 𝑀 = ( invg ‘ 𝑅 )
5		lcdneg.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
6		lcdneg.s	⊢ 𝑆 = ( Scalar ‘ 𝐶 )
7		lcdneg.n	⊢ 𝑁 = ( invg ‘ 𝑆 )
8		lcdneg.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		eqid	⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 )
10	1 2 3 9 5 6 8	lcdsca	⊢ ( 𝜑 → 𝑆 = ( oppr ‘ 𝑅 ) )
11	10	fveq2d	⊢ ( 𝜑 → ( invg ‘ 𝑆 ) = ( invg ‘ ( oppr ‘ 𝑅 ) ) )
12	9 4	opprneg	⊢ 𝑀 = ( invg ‘ ( oppr ‘ 𝑅 ) )
13	11 7 12	3eqtr4g	⊢ ( 𝜑 → 𝑁 = 𝑀 )