Metamath Proof Explorer

Theorem cdleme3c

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme3fa and cdleme3 . (Contributed by NM, 6-Jun-2012)

Ref Expression
Hypotheses cdleme1.l 
|- .<_ = ( le ` K )
cdleme1.j 
|- .\/ = ( join ` K )
cdleme1.m 
|- ./\ = ( meet ` K )
cdleme1.a 
|- A = ( Atoms ` K )
cdleme1.h 
|- H = ( LHyp ` K )
cdleme1.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme1.f 
|- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
cdleme3c.z 
|- .0. = ( 0. ` K )
Assertion cdleme3c 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F =/= .0. )

Proof

Step Hyp Ref Expression
1 cdleme1.l 
 |-  .<_ = ( le ` K )
2 cdleme1.j 
 |-  .\/ = ( join ` K )
3 cdleme1.m 
 |-  ./\ = ( meet ` K )
4 cdleme1.a 
 |-  A = ( Atoms ` K )
5 cdleme1.h 
 |-  H = ( LHyp ` K )
6 cdleme1.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme1.f 
 |-  F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
8 cdleme3c.z 
 |-  .0. = ( 0. ` K )
9 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. HL )
10 hllat 
 |-  ( K e. HL -> K e. Lat )
11 10 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. Lat )
12 simpr3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. A )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
15 12 14 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. ( Base ` K ) )
16 hlop 
 |-  ( K e. HL -> K e. OP )
17 16 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. OP )
18 13 8 op0cl 
 |-  ( K e. OP -> .0. e. ( Base ` K ) )
19 17 18 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> .0. e. ( Base ` K ) )
20 13 2 latjcl 
 |-  ( ( K e. Lat /\ R e. ( Base ` K ) /\ .0. e. ( Base ` K ) ) -> ( R .\/ .0. ) e. ( Base ` K ) )
21 11 15 19 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ .0. ) e. ( Base ` K ) )
22 simpl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
23 simpr1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. A )
24 simpr2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. A )
25 1 2 3 4 5 6 7 13 cdleme1b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> F e. ( Base ` K ) )
26 22 23 24 12 25 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F e. ( Base ` K ) )
27 13 2 latjcl 
 |-  ( ( K e. Lat /\ R e. ( Base ` K ) /\ F e. ( Base ` K ) ) -> ( R .\/ F ) e. ( Base ` K ) )
28 11 15 26 27 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) e. ( Base ` K ) )
29 13 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
30 23 29 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. ( Base ` K ) )
31 13 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
32 24 31 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. ( Base ` K ) )
33 13 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
34 11 30 32 33 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
35 13 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
36 35 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> W e. ( Base ` K ) )
37 13 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
38 11 34 36 37 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
39 6 38 eqbrtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U .<_ W )
40 simpr3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> -. R .<_ W )
41 nbrne2 
 |-  ( ( U .<_ W /\ -. R .<_ W ) -> U =/= R )
42 39 40 41 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U =/= R )
43 42 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R =/= U )
44 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 )
45 44 3adant3r3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U e. A )
46 eqid 
 |-  (
47 2 46 4 atcvr1 
 |-  ( ( K e. HL /\ R e. A /\ U e. A ) -> ( R =/= U <-> R (
48 9 12 45 47 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R =/= U <-> R (
49 43 48 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R (
50 hlol 
 |-  ( K e. HL -> K e. OL )
51 50 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. OL )
52 13 2 8 olj01 
 |-  ( ( K e. OL /\ R e. ( Base ` K ) ) -> ( R .\/ .0. ) = R )
53 51 15 52 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ .0. ) = R )
54 simpr3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R e. A /\ -. R .<_ W ) )
55 1 2 3 4 5 6 7 cdleme1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) = ( R .\/ U ) )
56 22 23 24 54 55 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) = ( R .\/ U ) )
57 49 53 56 3brtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ .0. ) (
58 13 46 cvrne 
 |-  ( ( ( K e. HL /\ ( R .\/ .0. ) e. ( Base ` K ) /\ ( R .\/ F ) e. ( Base ` K ) ) /\ ( R .\/ .0. ) (  ( R .\/ .0. ) =/= ( R .\/ F ) )
59 9 21 28 57 58 syl31anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ .0. ) =/= ( R .\/ F ) )
60 oveq2 
 |-  ( .0. = F -> ( R .\/ .0. ) = ( R .\/ F ) )
61 60 necon3i 
 |-  ( ( R .\/ .0. ) =/= ( R .\/ F ) -> .0. =/= F )
62 59 61 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> .0. =/= F )
63 62 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F =/= .0. )