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Metamath Proof Explorer

Theorem lhp2at0

Description: Join and meet with different atoms under co-atom W . (Contributed by NM, 15-Jun-2013)

Ref Expression
Hypotheses lhp2at0.l 
|- .<_ = ( le ` K )
lhp2at0.j 
|- .\/ = ( join ` K )
lhp2at0.m 
|- ./\ = ( meet ` K )
lhp2at0.z 
|- .0. = ( 0. ` K )
lhp2at0.a 
|- A = ( Atoms ` K )
lhp2at0.h 
|- H = ( LHyp ` K )
Assertion 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 ) ./\ V ) = .0. )

Proof

Step Hyp Ref Expression
1 lhp2at0.l 
 |-  .<_ = ( le ` K )
2 lhp2at0.j 
 |-  .\/ = ( join ` K )
3 lhp2at0.m 
 |-  ./\ = ( meet ` K )
4 lhp2at0.z 
 |-  .0. = ( 0. ` K )
5 lhp2at0.a 
 |-  A = ( Atoms ` K )
6 lhp2at0.h 
 |-  H = ( LHyp ` K )
7 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 )
8 hlol 
 |-  ( K e. HL -> K e. OL )
9 7 8 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. OL )
10 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 )
11 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 )
12 eqid 
 |-  ( Base ` K ) = ( Base ` K )
13 12 2 5 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ U e. A ) -> ( P .\/ U ) e. ( Base ` K ) )
14 7 10 11 13 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 ) )
15 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> W e. H )
16 12 6 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
17 15 16 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> W e. ( Base ` K ) )
18 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 )
19 12 5 atbase 
 |-  ( V e. A -> V e. ( Base ` K ) )
20 18 19 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 ) )
21 12 3 latmassOLD 
 |-  ( ( K e. OL /\ ( ( P .\/ U ) e. ( Base ` K ) /\ W e. ( Base ` K ) /\ V e. ( Base ` K ) ) ) -> ( ( ( P .\/ U ) ./\ W ) ./\ V ) = ( ( P .\/ U ) ./\ ( W ./\ V ) ) )
22 9 14 17 20 21 syl13anc 
 |-  ( ( ( ( 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 ) ./\ W ) ./\ V ) = ( ( P .\/ U ) ./\ ( W ./\ V ) ) )
23 1 3 4 5 6 lhpmat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = .0. )
24 23 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) -> ( P ./\ W ) = .0. )
25 24 3ad2ant1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( P ./\ W ) = .0. )
26 25 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( ( P ./\ W ) .\/ U ) = ( .0. .\/ U ) )
27 12 5 atbase 
 |-  ( U e. A -> U e. ( Base ` K ) )
28 11 27 syl 
 |-  ( ( ( ( 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. ( Base ` K ) )
29 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> U .<_ W )
30 12 1 2 3 5 atmod4i2 
 |-  ( ( K e. HL /\ ( P e. A /\ U e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ U .<_ W ) -> ( ( P ./\ W ) .\/ U ) = ( ( P .\/ U ) ./\ W ) )
31 7 10 28 17 29 30 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( ( P ./\ W ) .\/ U ) = ( ( P .\/ U ) ./\ W ) )
32 12 2 4 olj02 
 |-  ( ( K e. OL /\ U e. ( Base ` K ) ) -> ( .0. .\/ U ) = U )
33 9 28 32 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( .0. .\/ U ) = U )
34 26 31 33 3eqtr3d 
 |-  ( ( ( ( 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 ) ./\ W ) = U )
35 34 oveq1d 
 |-  ( ( ( ( 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 ) ./\ W ) ./\ V ) = ( U ./\ V ) )
36 22 35 eqtr3d 
 |-  ( ( ( ( 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 ) ./\ ( W ./\ V ) ) = ( U ./\ V ) )
37 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> V .<_ W )
38 7 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 )
39 12 1 3 latleeqm2 
 |-  ( ( K e. Lat /\ V e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( V .<_ W <-> ( W ./\ V ) = V ) )
40 38 20 17 39 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 .<_ W <-> ( W ./\ V ) = V ) )
41 37 40 mpbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( W ./\ V ) = V )
42 41 oveq2d 
 |-  ( ( ( ( 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 ) ./\ ( W ./\ V ) ) = ( ( P .\/ U ) ./\ V ) )
43 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> U =/= V )
44 hlatl 
 |-  ( K e. HL -> K e. AtLat )
45 7 44 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 )
46 3 4 5 atnem0 
 |-  ( ( K e. AtLat /\ U e. A /\ V e. A ) -> ( U =/= V <-> ( U ./\ V ) = .0. ) )
47 45 11 18 46 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( U =/= V <-> ( U ./\ V ) = .0. ) )
48 43 47 mpbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( U ./\ V ) = .0. )
49 36 42 48 3eqtr3d 
 |-  ( ( ( ( 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 ) ./\ V ) = .0. )