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	|- ( ph -> ( K e. HL /\ W e. H ) )
Assertion	lcdneg	|- ( ph -> 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	|- ( ph -> ( K e. HL /\ W e. H ) )
9		eqid	|- ( oppR ` R ) = ( oppR ` R )
10	1 2 3 9 5 6 8	lcdsca	|- ( ph -> S = ( oppR ` R ) )
11	10	fveq2d	|- ( ph -> ( invg ` S ) = ( invg ` ( oppR ` R ) ) )
12	9 4	opprneg	|- M = ( invg ` ( oppR ` R ) )
13	11 7 12	3eqtr4g	|- ( ph -> N = M )