Theorem lcd0

Description: The zero scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015)

	Ref	Expression
Hypotheses	lcd0.h	|- H = ( LHyp ` K )
	lcd0.u	|- U = ( ( DVecH ` K ) ` W )
	lcd0.f	|- F = ( Scalar ` U )
	lcd0.z	|- .0. = ( 0g ` F )
	lcd0.c	|- C = ( ( LCDual ` K ) ` W )
	lcd0.s	|- S = ( Scalar ` C )
	lcd0.o	|- O = ( 0g ` S )
	lcd0.k	|- ( ph -> ( K e. HL /\ W e. H ) )
Assertion	lcd0	|- ( ph -> O = .0. )

Proof

Step	Hyp	Ref	Expression
1		lcd0.h	|- H = ( LHyp ` K )
2		lcd0.u	|- U = ( ( DVecH ` K ) ` W )
3		lcd0.f	|- F = ( Scalar ` U )
4		lcd0.z	|- .0. = ( 0g ` F )
5		lcd0.c	|- C = ( ( LCDual ` K ) ` W )
6		lcd0.s	|- S = ( Scalar ` C )
7		lcd0.o	|- O = ( 0g ` S )
8		lcd0.k	|- ( ph -> ( K e. HL /\ W e. H ) )
9		eqid	|- ( oppR ` F ) = ( oppR ` F )
10	1 2 3 9 5 6 8	lcdsca	|- ( ph -> S = ( oppR ` F ) )
11	10	fveq2d	|- ( ph -> ( 0g ` S ) = ( 0g ` ( oppR ` F ) ) )
12	9 4	oppr0	|- .0. = ( 0g ` ( oppR ` F ) )
13	11 7 12	3eqtr4g	|- ( ph -> O = .0. )