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Metamath Proof Explorer

Theorem lhpexlt

Description: There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012)

Ref Expression
Hypotheses lhpatltex.s 
|- .< = ( lt ` K )
lhpatltex.a 
|- A = ( Atoms ` K )
lhpatltex.h 
|- H = ( LHyp ` K )
Assertion lhpexlt 
|- ( ( K e. HL /\ W e. H ) -> E. p e. A p .< W )

Proof

Step Hyp Ref Expression
1 lhpatltex.s 
 |-  .< = ( lt ` K )
2 lhpatltex.a 
 |-  A = ( Atoms ` K )
3 lhpatltex.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 
 |-  ( 1. ` K ) = ( 1. ` K )
9 eqid 
 |-  (
10 8 9 3 lhp1cvr 
 |-  ( ( K e. HL /\ W e. H ) -> W (
11 5 1 8 9 2 1cvratex 
 |-  ( ( K e. HL /\ W e. ( Base ` K ) /\ W (  E. p e. A p .< W )
12 4 7 10 11 syl3anc 
 |-  ( ( K e. HL /\ W e. H ) -> E. p e. A p .< W )