Theorem dochkrshp3

Description: Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015)

	Ref	Expression
Hypotheses	dochkrshp3.h	|- H = ( LHyp ` K )
	dochkrshp3.o	|- ._|_ = ( ( ocH ` K ) ` W )
	dochkrshp3.u	|- U = ( ( DVecH ` K ) ` W )
	dochkrshp3.v	|- V = ( Base ` U )
	dochkrshp3.f	|- F = ( LFnl ` U )
	dochkrshp3.l	|- L = ( LKer ` U )
	dochkrshp3.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	dochkrshp3.g	|- ( ph -> G e. F )
Assertion	dochkrshp3	|- ( ph -> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( L ` G ) =/= V ) ) )

Proof

Step	Hyp	Ref	Expression
1		dochkrshp3.h	|- H = ( LHyp ` K )
2		dochkrshp3.o	|- ._|_ = ( ( ocH ` K ) ` W )
3		dochkrshp3.u	|- U = ( ( DVecH ` K ) ` W )
4		dochkrshp3.v	|- V = ( Base ` U )
5		dochkrshp3.f	|- F = ( LFnl ` U )
6		dochkrshp3.l	|- L = ( LKer ` U )
7		dochkrshp3.k	|- ( ph -> ( K e. HL /\ W e. H ) )
8		dochkrshp3.g	|- ( ph -> G e. F )
9		eqid	|- ( LSHyp ` U ) = ( LSHyp ` U )
10	1 2 3 4 9 5 6 7 8	dochkrshp2	|- ( ph -> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( L ` G ) e. ( LSHyp ` U ) ) ) )
11	1 3 7	dvhlvec	|- ( ph -> U e. LVec )
12	4 9 5 6 11 8	lkrshp4	|- ( ph -> ( ( L ` G ) =/= V <-> ( L ` G ) e. ( LSHyp ` U ) ) )
13	12	anbi2d	|- ( ph -> ( ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( L ` G ) =/= V ) <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( L ` G ) e. ( LSHyp ` U ) ) ) )
14	10 13	bitr4d	|- ( ph -> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( L ` G ) =/= V ) ) )