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Metamath Proof Explorer

Theorem lhplt

Description: An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012)

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

Proof

Step Hyp Ref Expression
1 lhplt.l 
 |-  .<_ = ( le ` K )
2 lhplt.s 
 |-  .< = ( lt ` K )
3 lhplt.a 
 |-  A = ( Atoms ` K )
4 lhplt.h 
 |-  H = ( LHyp ` K )
5 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ P .<_ W ) ) -> K e. HL )
6 simprl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ P .<_ W ) ) -> P e. A )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 4 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
9 8 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ P .<_ W ) ) -> W e. ( Base ` K ) )
10 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
11 eqid 
 |-  (
12 10 11 4 lhp1cvr 
 |-  ( ( K e. HL /\ W e. H ) -> W (
13 12 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ P .<_ W ) ) -> W (
14 simprr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ P .<_ W ) ) -> P .<_ W )
15 7 1 2 10 11 3 1cvratlt 
 |-  ( ( ( K e. HL /\ P e. A /\ W e. ( Base ` K ) ) /\ ( W (  P .< W )
16 5 6 9 13 14 15 syl32anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ P .<_ W ) ) -> P .< W )