Theorem lcfl2

Description: Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015)

	Ref	Expression
Hypotheses	lcfl2.h	|- H = ( LHyp ` K )
	lcfl2.o	|- ._|_ = ( ( ocH ` K ) ` W )
	lcfl2.u	|- U = ( ( DVecH ` K ) ` W )
	lcfl2.v	|- V = ( Base ` U )
	lcfl2.f	|- F = ( LFnl ` U )
	lcfl2.l	|- L = ( LKer ` U )
	lcfl2.c	|- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
	lcfl2.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	lcfl2.g	|- ( ph -> G e. F )
Assertion	lcfl2	|- ( ph -> ( G e. C <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V \/ ( L ` G ) = V ) ) )

Proof

Step	Hyp	Ref	Expression
1		lcfl2.h	|- H = ( LHyp ` K )
2		lcfl2.o	|- ._|_ = ( ( ocH ` K ) ` W )
3		lcfl2.u	|- U = ( ( DVecH ` K ) ` W )
4		lcfl2.v	|- V = ( Base ` U )
5		lcfl2.f	|- F = ( LFnl ` U )
6		lcfl2.l	|- L = ( LKer ` U )
7		lcfl2.c	|- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
8		lcfl2.k	|- ( ph -> ( K e. HL /\ W e. H ) )
9		lcfl2.g	|- ( ph -> G e. F )
10	7 9	lcfl1	|- ( ph -> ( G e. C <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) )
11	1 2 3 4 5 6 8 9	dochkrshp4	|- ( ph -> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V \/ ( L ` G ) = V ) ) )
12	10 11	bitrd	|- ( ph -> ( G e. C <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V \/ ( L ` G ) = V ) ) )