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Metamath Proof Explorer

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 ) ) )