Metamath Proof Explorer

Theorem lhp2atnle

Description: Inequality for 2 different atoms under co-atom W . (Contributed by NM, 17-Jun-2013)

Ref Expression
Hypotheses lhp2atnle.l 
|- .<_ = ( le ` K )
lhp2atnle.j 
|- .\/ = ( join ` K )
lhp2atnle.a 
|- A = ( Atoms ` K )
lhp2atnle.h 
|- H = ( LHyp ` K )
Assertion lhp2atnle 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> -. V .<_ ( P .\/ U ) )

Proof

Step Hyp Ref Expression
1 lhp2atnle.l 
 |-  .<_ = ( le ` K )
2 lhp2atnle.j 
 |-  .\/ = ( join ` K )
3 lhp2atnle.a 
 |-  A = ( Atoms ` K )
4 lhp2atnle.h 
 |-  H = ( LHyp ` K )
5 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> K e. HL )
6 hlatl 
 |-  ( K e. HL -> K e. AtLat )
7 5 6 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> K e. AtLat )
8 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> V e. A )
9 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
10 9 3 atn0 
 |-  ( ( K e. AtLat /\ V e. A ) -> V =/= ( 0. ` K ) )
11 7 8 10 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> V =/= ( 0. ` K ) )
12 5 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> K e. Lat )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 3 atbase 
 |-  ( V e. A -> V e. ( Base ` K ) )
15 8 14 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> V e. ( Base ` K ) )
16 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> P e. A )
17 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> U e. A )
18 13 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ U e. A ) -> ( P .\/ U ) e. ( Base ` K ) )
19 5 16 17 18 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( P .\/ U ) e. ( Base ` K ) )
20 eqid 
 |-  ( meet ` K ) = ( meet ` K )
21 13 1 20 latleeqm2 
 |-  ( ( K e. Lat /\ V e. ( Base ` K ) /\ ( P .\/ U ) e. ( Base ` K ) ) -> ( V .<_ ( P .\/ U ) <-> ( ( P .\/ U ) ( meet ` K ) V ) = V ) )
22 12 15 19 21 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( V .<_ ( P .\/ U ) <-> ( ( P .\/ U ) ( meet ` K ) V ) = V ) )
23 1 2 20 9 3 4 lhp2at0 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( ( P .\/ U ) ( meet ` K ) V ) = ( 0. ` K ) )
24 eqeq1 
 |-  ( ( ( P .\/ U ) ( meet ` K ) V ) = V -> ( ( ( P .\/ U ) ( meet ` K ) V ) = ( 0. ` K ) <-> V = ( 0. ` K ) ) )
25 23 24 syl5ibcom 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( ( ( P .\/ U ) ( meet ` K ) V ) = V -> V = ( 0. ` K ) ) )
26 22 25 sylbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( V .<_ ( P .\/ U ) -> V = ( 0. ` K ) ) )
27 26 necon3ad 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( V =/= ( 0. ` K ) -> -. V .<_ ( P .\/ U ) ) )
28 11 27 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> -. V .<_ ( P .\/ U ) )