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Metamath Proof Explorer

Theorem lhpexle

Description: There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013)

Ref Expression
Hypotheses lhp2a.l 
|- .<_ = ( le ` K )
lhp2a.a 
|- A = ( Atoms ` K )
lhp2a.h 
|- H = ( LHyp ` K )
Assertion lhpexle 
|- ( ( K e. HL /\ W e. H ) -> E. p e. A p .<_ W )

Proof

Step Hyp Ref Expression
1 lhp2a.l 
 |-  .<_ = ( le ` K )
2 lhp2a.a 
 |-  A = ( Atoms ` K )
3 lhp2a.h 
 |-  H = ( LHyp ` K )
4 simpl 
 |-  ( ( K e. HL /\ W e. H ) -> K e. HL )
5 eqid 
 |-  ( Base ` K ) = ( Base ` K )
6 5 3 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
7 6 adantl 
 |-  ( ( K e. HL /\ W e. H ) -> W e. ( Base ` K ) )
8 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
9 8 3 lhpn0 
 |-  ( ( K e. HL /\ W e. H ) -> W =/= ( 0. ` K ) )
10 5 1 8 2 atle 
 |-  ( ( K e. HL /\ W e. ( Base ` K ) /\ W =/= ( 0. ` K ) ) -> E. p e. A p .<_ W )
11 4 7 9 10 syl3anc 
 |-  ( ( K e. HL /\ W e. H ) -> E. p e. A p .<_ W )