Metamath Proof Explorer

Theorem lcfl4N

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

Ref Expression
Hypotheses lcfl4.h 
|- H = ( LHyp ` K )
lcfl4.o 
|- ._|_ = ( ( ocH ` K ) ` W )
lcfl4.u 
|- U = ( ( DVecH ` K ) ` W )
lcfl4.v 
|- V = ( Base ` U )
lcfl4.y 
|- Y = ( LSHyp ` U )
lcfl4.f 
|- F = ( LFnl ` U )
lcfl4.l 
|- L = ( LKer ` U )
lcfl4.c 
|- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
lcfl4.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
lcfl4.g 
|- ( ph -> G e. F )
Assertion lcfl4N 
|- ( ph -> ( G e. C <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) e. Y \/ ( L ` G ) = V ) ) )

Proof

Step Hyp Ref Expression
1 lcfl4.h 
 |-  H = ( LHyp ` K )
2 lcfl4.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
3 lcfl4.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 lcfl4.v 
 |-  V = ( Base ` U )
5 lcfl4.y 
 |-  Y = ( LSHyp ` U )
6 lcfl4.f 
 |-  F = ( LFnl ` U )
7 lcfl4.l 
 |-  L = ( LKer ` U )
8 lcfl4.c 
 |-  C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
9 lcfl4.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
10 lcfl4.g 
 |-  ( ph -> G e. F )
11 eqid 
 |-  ( LSAtoms ` U ) = ( LSAtoms ` U )
12 1 2 3 4 11 6 7 8 9 10 lcfl3 
 |-  ( ph -> ( G e. C <-> ( ( ._|_ ` ( L ` G ) ) e. ( LSAtoms ` U ) \/ ( L ` G ) = V ) ) )
13 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
14 1 3 9 dvhlmod 
 |-  ( ph -> U e. LMod )
15 4 6 7 14 10 lkrssv 
 |-  ( ph -> ( L ` G ) C_ V )
16 1 3 4 13 2 dochlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( L ` G ) C_ V ) -> ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) )
17 9 15 16 syl2anc 
 |-  ( ph -> ( ._|_ ` ( L ` G ) ) e. ( LSubSp ` U ) )
18 1 2 3 13 11 5 9 17 dochsatshpb 
 |-  ( ph -> ( ( ._|_ ` ( L ` G ) ) e. ( LSAtoms ` U ) <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) e. Y ) )
19 18 orbi1d 
 |-  ( ph -> ( ( ( ._|_ ` ( L ` G ) ) e. ( LSAtoms ` U ) \/ ( L ` G ) = V ) <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) e. Y \/ ( L ` G ) = V ) ) )
20 12 19 bitrd 
 |-  ( ph -> ( G e. C <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) e. Y \/ ( L ` G ) = V ) ) )