Theorem lcd0v2

Description: The zero functional in the set of functionals with closed kernels. (Contributed by NM, 27-Mar-2015)

	Ref	Expression
Hypotheses	lcd0v2.h	|- H = ( LHyp ` K )
	lcd0v2.u	|- U = ( ( DVecH ` K ) ` W )
	lcd0v2.d	|- D = ( LDual ` U )
	lcd0v2.z	|- .0. = ( 0g ` D )
	lcd0v2.c	|- C = ( ( LCDual ` K ) ` W )
	lcd0v2.o	|- O = ( 0g ` C )
	lcd0v2.k	|- ( ph -> ( K e. HL /\ W e. H ) )
Assertion	lcd0v2	|- ( ph -> O = .0. )

Proof

Step	Hyp	Ref	Expression
1		lcd0v2.h	|- H = ( LHyp ` K )
2		lcd0v2.u	|- U = ( ( DVecH ` K ) ` W )
3		lcd0v2.d	|- D = ( LDual ` U )
4		lcd0v2.z	|- .0. = ( 0g ` D )
5		lcd0v2.c	|- C = ( ( LCDual ` K ) ` W )
6		lcd0v2.o	|- O = ( 0g ` C )
7		lcd0v2.k	|- ( ph -> ( K e. HL /\ W e. H ) )
8		eqid	|- ( Base ` U ) = ( Base ` U )
9		eqid	|- ( Scalar ` U ) = ( Scalar ` U )
10		eqid	|- ( 0g ` ( Scalar ` U ) ) = ( 0g ` ( Scalar ` U ) )
11	1 2 8 9 10 5 6 7	lcd0v	|- ( ph -> O = ( ( Base ` U ) X. { ( 0g ` ( Scalar ` U ) ) } ) )
12	1 2 7	dvhlmod	|- ( ph -> U e. LMod )
13	8 9 10 3 4 12	ldual0v	|- ( ph -> .0. = ( ( Base ` U ) X. { ( 0g ` ( Scalar ` U ) ) } ) )
14	11 13	eqtr4d	|- ( ph -> O = .0. )