Metamath Proof Explorer


Theorem cdleme0e

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 13-Jun-2012)

Ref Expression
Hypotheses cdleme0.l
|- .<_ = ( le ` K )
cdleme0.j
|- .\/ = ( join ` K )
cdleme0.m
|- ./\ = ( meet ` K )
cdleme0.a
|- A = ( Atoms ` K )
cdleme0.h
|- H = ( LHyp ` K )
cdleme0.u
|- U = ( ( P .\/ Q ) ./\ W )
cdleme0c.3
|- V = ( ( P .\/ R ) ./\ W )
Assertion cdleme0e
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> U =/= V )

Proof

Step Hyp Ref Expression
1 cdleme0.l
 |-  .<_ = ( le ` K )
2 cdleme0.j
 |-  .\/ = ( join ` K )
3 cdleme0.m
 |-  ./\ = ( meet ` K )
4 cdleme0.a
 |-  A = ( Atoms ` K )
5 cdleme0.h
 |-  H = ( LHyp ` K )
6 cdleme0.u
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme0c.3
 |-  V = ( ( P .\/ R ) ./\ W )
8 6 7 oveq12i
 |-  ( U ./\ V ) = ( ( ( P .\/ Q ) ./\ W ) ./\ ( ( P .\/ R ) ./\ W ) )
9 simp1l
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. HL )
10 hlol
 |-  ( K e. HL -> K e. OL )
11 9 10 syl
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. OL )
12 simp21l
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P e. A )
13 simp22
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q e. A )
14 eqid
 |-  ( Base ` K ) = ( Base ` K )
15 14 2 4 hlatjcl
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
16 9 12 13 15 syl3anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
17 simp23l
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> R e. A )
18 14 2 4 hlatjcl
 |-  ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) e. ( Base ` K ) )
19 9 12 17 18 syl3anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P .\/ R ) e. ( Base ` K ) )
20 simp1r
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> W e. H )
21 14 5 lhpbase
 |-  ( W e. H -> W e. ( Base ` K ) )
22 20 21 syl
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> W e. ( Base ` K ) )
23 14 3 latmmdir
 |-  ( ( K e. OL /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ ( P .\/ R ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) ./\ ( P .\/ R ) ) ./\ W ) = ( ( ( P .\/ Q ) ./\ W ) ./\ ( ( P .\/ R ) ./\ W ) ) )
24 11 16 19 22 23 syl13anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( ( ( P .\/ Q ) ./\ ( P .\/ R ) ) ./\ W ) = ( ( ( P .\/ Q ) ./\ W ) ./\ ( ( P .\/ R ) ./\ W ) ) )
25 9 hllatd
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. Lat )
26 14 4 atbase
 |-  ( R e. A -> R e. ( Base ` K ) )
27 17 26 syl
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> R e. ( Base ` K ) )
28 14 4 atbase
 |-  ( P e. A -> P e. ( Base ` K ) )
29 12 28 syl
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P e. ( Base ` K ) )
30 14 4 atbase
 |-  ( Q e. A -> Q e. ( Base ` K ) )
31 13 30 syl
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q e. ( Base ` K ) )
32 simp3r
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. R .<_ ( P .\/ Q ) )
33 14 1 2 latnlej1r
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= Q )
34 33 necomd
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> Q =/= R )
35 25 27 29 31 32 34 syl131anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q =/= R )
36 simp3
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) )
37 1 2 4 hlatcon3
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. P .<_ ( Q .\/ R ) )
38 9 12 13 17 36 37 syl131anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. P .<_ ( Q .\/ R ) )
39 1 2 3 4 2llnma2
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( Q =/= R /\ -. P .<_ ( Q .\/ R ) ) ) -> ( ( P .\/ Q ) ./\ ( P .\/ R ) ) = P )
40 9 13 17 12 35 38 39 syl132anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( P .\/ R ) ) = P )
41 40 oveq1d
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( ( ( P .\/ Q ) ./\ ( P .\/ R ) ) ./\ W ) = ( P ./\ W ) )
42 24 41 eqtr3d
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( ( ( P .\/ Q ) ./\ W ) ./\ ( ( P .\/ R ) ./\ W ) ) = ( P ./\ W ) )
43 8 42 syl5eq
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( U ./\ V ) = ( P ./\ W ) )
44 simp1
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
45 simp21
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
46 eqid
 |-  ( 0. ` K ) = ( 0. ` K )
47 1 3 46 4 5 lhpmat
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = ( 0. ` K ) )
48 44 45 47 syl2anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P ./\ W ) = ( 0. ` K ) )
49 43 48 eqtrd
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( U ./\ V ) = ( 0. ` K ) )
50 hlatl
 |-  ( K e. HL -> K e. AtLat )
51 9 50 syl
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. AtLat )
52 simp3l
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P =/= Q )
53 1 2 3 4 5 6 lhpat2
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
54 44 45 13 52 53 syl112anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> U e. A )
55 14 1 2 latnlej1l
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= P )
56 55 necomd
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> P =/= R )
57 25 27 29 31 32 56 syl131anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P =/= R )
58 1 2 3 4 5 7 lhpat2
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ P =/= R ) ) -> V e. A )
59 44 45 17 57 58 syl112anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> V e. A )
60 3 46 4 atnem0
 |-  ( ( K e. AtLat /\ U e. A /\ V e. A ) -> ( U =/= V <-> ( U ./\ V ) = ( 0. ` K ) ) )
61 51 54 59 60 syl3anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( U =/= V <-> ( U ./\ V ) = ( 0. ` K ) ) )
62 49 61 mpbird
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> U =/= V )