Metamath Proof Explorer

Theorem lhpjat1

Description: The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012)

Ref Expression
Hypotheses lhpjat.l 
|- .<_ = ( le ` K )
lhpjat.j 
|- .\/ = ( join ` K )
lhpjat.u 
|- .1. = ( 1. ` K )
lhpjat.a 
|- A = ( Atoms ` K )
lhpjat.h 
|- H = ( LHyp ` K )
Assertion lhpjat1 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( W .\/ P ) = .1. )

Proof

Step Hyp Ref Expression
1 lhpjat.l 
 |-  .<_ = ( le ` K )
2 lhpjat.j 
 |-  .\/ = ( join ` K )
3 lhpjat.u 
 |-  .1. = ( 1. ` K )
4 lhpjat.a 
 |-  A = ( Atoms ` K )
5 lhpjat.h 
 |-  H = ( LHyp ` K )
6 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> K e. HL )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 5 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 simprl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> P e. A )
11 eqid 
 |-  (
12 3 11 5 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 3 11 4 1cvrjat 
 |-  ( ( ( K e. HL /\ W e. ( Base ` K ) /\ P e. A ) /\ ( W (  ( W .\/ P ) = .1. )
16 6 9 10 13 14 15 syl32anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( W .\/ P ) = .1. )