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Metamath Proof Explorer

Theorem lhpjat2

Description: The join of a co-atom (hyperplane) and an atom not under it is the lattice unity. (Contributed by NM, 4-Jun-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 lhpjat2 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ W ) = .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 hllat 
 |-  ( K e. HL -> K e. Lat )
7 6 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> K e. Lat )
8 eqid 
 |-  ( Base ` K ) = ( Base ` K )
9 8 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
10 9 ad2antrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> P e. ( Base ` K ) )
11 8 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
12 11 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> W e. ( Base ` K ) )
13 8 2 latjcom 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( P .\/ W ) = ( W .\/ P ) )
14 7 10 12 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ W ) = ( W .\/ P ) )
15 1 2 3 4 5 lhpjat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( W .\/ P ) = .1. )
16 14 15 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ W ) = .1. )