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Metamath Proof Explorer

Theorem lhpexnle

Description: There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013)

Ref Expression
Hypotheses lhp2a.l 
|- .<_ = ( le ` K )
lhp2a.a 
|- A = ( Atoms ` K )
lhp2a.h 
|- H = ( LHyp ` K )
Assertion lhpexnle 
|- ( ( 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 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
5 eqid 
 |-  (
6 4 5 3 lhp1cvr 
 |-  ( ( K e. HL /\ W e. H ) -> W (
7 simpl 
 |-  ( ( K e. HL /\ W e. H ) -> K e. HL )
8 eqid 
 |-  ( Base ` K ) = ( Base ` K )
9 8 3 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
10 9 adantl 
 |-  ( ( K e. HL /\ W e. H ) -> W e. ( Base ` K ) )
11 hlop 
 |-  ( K e. HL -> K e. OP )
12 8 4 op1cl 
 |-  ( K e. OP -> ( 1. ` K ) e. ( Base ` K ) )
13 11 12 syl 
 |-  ( K e. HL -> ( 1. ` K ) e. ( Base ` K ) )
14 13 adantr 
 |-  ( ( K e. HL /\ W e. H ) -> ( 1. ` K ) e. ( Base ` K ) )
15 eqid 
 |-  ( join ` K ) = ( join ` K )
16 8 1 15 5 2 cvrval3 
 |-  ( ( K e. HL /\ W e. ( Base ` K ) /\ ( 1. ` K ) e. ( Base ` K ) ) -> ( W (  E. p e. A ( -. p .<_ W /\ ( W ( join ` K ) p ) = ( 1. ` K ) ) ) )
17 7 10 14 16 syl3anc 
 |-  ( ( K e. HL /\ W e. H ) -> ( W (  E. p e. A ( -. p .<_ W /\ ( W ( join ` K ) p ) = ( 1. ` K ) ) ) )
18 6 17 mpbid 
 |-  ( ( K e. HL /\ W e. H ) -> E. p e. A ( -. p .<_ W /\ ( W ( join ` K ) p ) = ( 1. ` K ) ) )
19 simpl 
 |-  ( ( -. p .<_ W /\ ( W ( join ` K ) p ) = ( 1. ` K ) ) -> -. p .<_ W )
20 19 reximi 
 |-  ( E. p e. A ( -. p .<_ W /\ ( W ( join ` K ) p ) = ( 1. ` K ) ) -> E. p e. A -. p .<_ W )
21 18 20 syl 
 |-  ( ( K e. HL /\ W e. H ) -> E. p e. A -. p .<_ W )