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Metamath Proof Explorer

Theorem dochkrsat2

Description: The orthocomplement of a kernel is an atom iff the double orthocomplement is not the vector space. (Contributed by NM, 1-Jan-2015)

Ref Expression
Hypotheses dochkrsat2.h 
|- H = ( LHyp ` K )
dochkrsat2.o 
|- ._|_ = ( ( ocH ` K ) ` W )
dochkrsat2.u 
|- U = ( ( DVecH ` K ) ` W )
dochkrsat2.v 
|- V = ( Base ` U )
dochkrsat2.a 
|- A = ( LSAtoms ` U )
dochkrsat2.f 
|- F = ( LFnl ` U )
dochkrsat2.l 
|- L = ( LKer ` U )
dochkrsat2.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dochkrsat2.g 
|- ( ph -> G e. F )
Assertion dochkrsat2 
|- ( ph -> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V <-> ( ._|_ ` ( L ` G ) ) e. A ) )

Proof

Step Hyp Ref Expression
1 dochkrsat2.h 
 |-  H = ( LHyp ` K )
2 dochkrsat2.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
3 dochkrsat2.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 dochkrsat2.v 
 |-  V = ( Base ` U )
5 dochkrsat2.a 
 |-  A = ( LSAtoms ` U )
6 dochkrsat2.f 
 |-  F = ( LFnl ` U )
7 dochkrsat2.l 
 |-  L = ( LKer ` U )
8 dochkrsat2.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
9 dochkrsat2.g 
 |-  ( ph -> G e. F )
10 eqid 
 |-  ( 0g ` U ) = ( 0g ` U )
11 1 3 8 dvhlmod 
 |-  ( ph -> U e. LMod )
12 4 6 7 11 9 lkrssv 
 |-  ( ph -> ( L ` G ) C_ V )
13 1 2 3 4 10 8 12 dochn0nv 
 |-  ( ph -> ( ( ._|_ ` ( L ` G ) ) =/= { ( 0g ` U ) } <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V ) )
14 1 2 3 5 6 7 10 8 9 dochkrsat 
 |-  ( ph -> ( ( ._|_ ` ( L ` G ) ) =/= { ( 0g ` U ) } <-> ( ._|_ ` ( L ` G ) ) e. A ) )
15 13 14 bitr3d 
 |-  ( ph -> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) =/= V <-> ( ._|_ ` ( L ` G ) ) e. A ) )