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Metamath Proof Explorer

Theorem lhpj1

Description: The join of a co-atom (hyperplane) and an element not under it is the lattice unity. (Contributed by NM, 7-Dec-2012)

Ref Expression
Hypotheses lhpj1.b 
|- B = ( Base ` K )
lhpj1.l 
|- .<_ = ( le ` K )
lhpj1.j 
|- .\/ = ( join ` K )
lhpj1.u 
|- .1. = ( 1. ` K )
lhpj1.h 
|- H = ( LHyp ` K )
Assertion lhpj1 
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( W .\/ X ) = .1. )

Proof

Step Hyp Ref Expression
1 lhpj1.b 
 |-  B = ( Base ` K )
2 lhpj1.l 
 |-  .<_ = ( le ` K )
3 lhpj1.j 
 |-  .\/ = ( join ` K )
4 lhpj1.u 
 |-  .1. = ( 1. ` K )
5 lhpj1.h 
 |-  H = ( LHyp ` K )
6 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> K e. HL )
7 simpr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> X e. B )
8 1 5 lhpbase 
 |-  ( W e. H -> W e. B )
9 8 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> W e. B )
10 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
11 1 2 10 hlrelat2 
 |-  ( ( K e. HL /\ X e. B /\ W e. B ) -> ( -. X .<_ W <-> E. p e. ( Atoms ` K ) ( p .<_ X /\ -. p .<_ W ) ) )
12 6 7 9 11 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( -. X .<_ W <-> E. p e. ( Atoms ` K ) ( p .<_ X /\ -. p .<_ W ) ) )
13 simp1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( K e. HL /\ W e. H ) )
14 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> p e. ( Atoms ` K ) )
15 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> -. p .<_ W )
16 2 3 4 10 5 lhpjat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( p e. ( Atoms ` K ) /\ -. p .<_ W ) ) -> ( W .\/ p ) = .1. )
17 13 14 15 16 syl12anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( W .\/ p ) = .1. )
18 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> p .<_ X )
19 simp1ll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> K e. HL )
20 19 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> K e. Lat )
21 1 10 atbase 
 |-  ( p e. ( Atoms ` K ) -> p e. B )
22 21 3ad2ant2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> p e. B )
23 simp1r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> X e. B )
24 9 3ad2ant1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> W e. B )
25 1 2 3 latjlej2 
 |-  ( ( K e. Lat /\ ( p e. B /\ X e. B /\ W e. B ) ) -> ( p .<_ X -> ( W .\/ p ) .<_ ( W .\/ X ) ) )
26 20 22 23 24 25 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( p .<_ X -> ( W .\/ p ) .<_ ( W .\/ X ) ) )
27 18 26 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( W .\/ p ) .<_ ( W .\/ X ) )
28 17 27 eqbrtrrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> .1. .<_ ( W .\/ X ) )
29 hlop 
 |-  ( K e. HL -> K e. OP )
30 19 29 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> K e. OP )
31 1 3 latjcl 
 |-  ( ( K e. Lat /\ W e. B /\ X e. B ) -> ( W .\/ X ) e. B )
32 20 24 23 31 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( W .\/ X ) e. B )
33 1 2 4 op1le 
 |-  ( ( K e. OP /\ ( W .\/ X ) e. B ) -> ( .1. .<_ ( W .\/ X ) <-> ( W .\/ X ) = .1. ) )
34 30 32 33 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( .1. .<_ ( W .\/ X ) <-> ( W .\/ X ) = .1. ) )
35 28 34 mpbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ p e. ( Atoms ` K ) /\ ( p .<_ X /\ -. p .<_ W ) ) -> ( W .\/ X ) = .1. )
36 35 rexlimdv3a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( E. p e. ( Atoms ` K ) ( p .<_ X /\ -. p .<_ W ) -> ( W .\/ X ) = .1. ) )
37 12 36 sylbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( -. X .<_ W -> ( W .\/ X ) = .1. ) )
38 37 impr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( W .\/ X ) = .1. )