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Metamath Proof Explorer

Theorem cdlemc6

Description: Lemma for cdlemc . (Contributed by NM, 26-May-2012)

Ref Expression
Hypotheses cdlemc3.l 
|- .<_ = ( le ` K )
cdlemc3.j 
|- .\/ = ( join ` K )
cdlemc3.m 
|- ./\ = ( meet ` K )
cdlemc3.a 
|- A = ( Atoms ` K )
cdlemc3.h 
|- H = ( LHyp ` K )
cdlemc3.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemc3.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemc6 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( F ` Q ) = ( ( Q .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemc3.l 
 |-  .<_ = ( le ` K )
2 cdlemc3.j 
 |-  .\/ = ( join ` K )
3 cdlemc3.m 
 |-  ./\ = ( meet ` K )
4 cdlemc3.a 
 |-  A = ( Atoms ` K )
5 cdlemc3.h 
 |-  H = ( LHyp ` K )
6 cdlemc3.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemc3.r 
 |-  R = ( ( trL ` K ) ` W )
8 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> K e. HL )
9 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> P e. A )
10 simp23l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> Q e. A )
11 2 4 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
12 8 9 10 11 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( P .\/ Q ) = ( Q .\/ P ) )
13 12 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( Q ./\ ( P .\/ Q ) ) = ( Q ./\ ( Q .\/ P ) ) )
14 8 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> K e. Lat )
15 eqid 
 |-  ( Base ` K ) = ( Base ` K )
16 15 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
17 10 16 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> Q e. ( Base ` K ) )
18 15 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
19 9 18 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> P e. ( Base ` K ) )
20 15 2 3 latabs2 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ P e. ( Base ` K ) ) -> ( Q ./\ ( Q .\/ P ) ) = Q )
21 14 17 19 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( Q ./\ ( Q .\/ P ) ) = Q )
22 13 21 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( Q ./\ ( P .\/ Q ) ) = Q )
23 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( K e. HL /\ W e. H ) )
24 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( P e. A /\ -. P .<_ W ) )
25 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> F e. T )
26 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( F ` P ) = P )
27 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
28 1 27 4 5 6 7 trl0 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = ( 0. ` K ) )
29 23 24 25 26 28 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( R ` F ) = ( 0. ` K ) )
30 29 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( R ` F ) ) = ( Q .\/ ( 0. ` K ) ) )
31 hlol 
 |-  ( K e. HL -> K e. OL )
32 8 31 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> K e. OL )
33 15 2 27 olj01 
 |-  ( ( K e. OL /\ Q e. ( Base ` K ) ) -> ( Q .\/ ( 0. ` K ) ) = Q )
34 32 17 33 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( 0. ` K ) ) = Q )
35 30 34 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( R ` F ) ) = Q )
36 26 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) = ( P .\/ ( ( P .\/ Q ) ./\ W ) ) )
37 15 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
38 8 9 10 37 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( P .\/ Q ) e. ( Base ` K ) )
39 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> W e. H )
40 15 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
41 39 40 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> W e. ( Base ` K ) )
42 15 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) e. ( Base ` K ) )
43 14 38 41 42 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( ( P .\/ Q ) ./\ W ) e. ( Base ` K ) )
44 15 2 latjcom 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( ( P .\/ Q ) ./\ W ) e. ( Base ` K ) ) -> ( P .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( ( P .\/ Q ) ./\ W ) .\/ P ) )
45 14 19 43 44 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( P .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( ( P .\/ Q ) ./\ W ) .\/ P ) )
46 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> P .<_ ( P .\/ Q ) )
47 8 9 10 46 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> P .<_ ( P .\/ Q ) )
48 15 1 2 3 4 atmod2i1 
 |-  ( ( K e. HL /\ ( P e. A /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ P .<_ ( P .\/ Q ) ) -> ( ( ( P .\/ Q ) ./\ W ) .\/ P ) = ( ( P .\/ Q ) ./\ ( W .\/ P ) ) )
49 8 9 38 41 47 48 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( ( ( P .\/ Q ) ./\ W ) .\/ P ) = ( ( P .\/ Q ) ./\ ( W .\/ P ) ) )
50 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
51 1 2 50 4 5 lhpjat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( W .\/ P ) = ( 1. ` K ) )
52 8 39 24 51 syl21anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( W .\/ P ) = ( 1. ` K ) )
53 52 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( ( P .\/ Q ) ./\ ( W .\/ P ) ) = ( ( P .\/ Q ) ./\ ( 1. ` K ) ) )
54 15 3 50 olm11 
 |-  ( ( K e. OL /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
55 32 38 54 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
56 49 53 55 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( ( ( P .\/ Q ) ./\ W ) .\/ P ) = ( P .\/ Q ) )
57 36 45 56 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) = ( P .\/ Q ) )
58 35 57 oveq12d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( ( Q .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) = ( Q ./\ ( P .\/ Q ) ) )
59 1 4 5 6 ltrnateq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( F ` Q ) = Q )
60 22 58 59 3eqtr4rd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = P ) -> ( F ` Q ) = ( ( Q .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) )