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Metamath Proof Explorer

Theorem lhpocnel2

Description: The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 20-Feb-2014)

Ref Expression
Hypotheses lhpocnel2.l 
|- .<_ = ( le ` K )
lhpocnel2.a 
|- A = ( Atoms ` K )
lhpocnel2.h 
|- H = ( LHyp ` K )
lhpocnel2.p 
|- P = ( ( oc ` K ) ` W )
Assertion lhpocnel2 
|- ( ( K e. HL /\ W e. H ) -> ( P e. A /\ -. P .<_ W ) )

Proof

Step Hyp Ref Expression
1 lhpocnel2.l 
 |-  .<_ = ( le ` K )
2 lhpocnel2.a 
 |-  A = ( Atoms ` K )
3 lhpocnel2.h 
 |-  H = ( LHyp ` K )
4 lhpocnel2.p 
 |-  P = ( ( oc ` K ) ` W )
5 eqid 
 |-  ( oc ` K ) = ( oc ` K )
6 1 5 2 3 lhpocnel 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) )
7 4 eleq1i 
 |-  ( P e. A <-> ( ( oc ` K ) ` W ) e. A )
8 4 breq1i 
 |-  ( P .<_ W <-> ( ( oc ` K ) ` W ) .<_ W )
9 8 notbii 
 |-  ( -. P .<_ W <-> -. ( ( oc ` K ) ` W ) .<_ W )
10 7 9 anbi12i 
 |-  ( ( P e. A /\ -. P .<_ W ) <-> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) )
11 6 10 sylibr 
 |-  ( ( K e. HL /\ W e. H ) -> ( P e. A /\ -. P .<_ W ) )