Metamath Proof Explorer

Theorem trljat2

Description: The value of a translation of an atom P not under the fiducial co-atom W , joined with trace. Equation above Lemma C in Crawley p. 112. (Contributed by NM, 25-May-2012)

Ref Expression
Hypotheses trljat.l 
|- .<_ = ( le ` K )
trljat.j 
|- .\/ = ( join ` K )
trljat.a 
|- A = ( Atoms ` K )
trljat.h 
|- H = ( LHyp ` K )
trljat.t 
|- T = ( ( LTrn ` K ) ` W )
trljat.r 
|- R = ( ( trL ` K ) ` W )
Assertion trljat2 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` P ) .\/ ( R ` F ) ) = ( P .\/ ( F ` P ) ) )

Proof

Step Hyp Ref Expression
1 trljat.l 
 |-  .<_ = ( le ` K )
2 trljat.j 
 |-  .\/ = ( join ` K )
3 trljat.a 
 |-  A = ( Atoms ` K )
4 trljat.h 
 |-  H = ( LHyp ` K )
5 trljat.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 trljat.r 
 |-  R = ( ( trL ` K ) ` W )
7 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> K e. HL )
8 1 3 4 5 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A )
9 8 3adant3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) e. A )
10 7 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> K e. Lat )
11 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> P e. A )
12 eqid 
 |-  ( Base ` K ) = ( Base ` K )
13 12 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
14 11 13 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> P e. ( Base ` K ) )
15 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) )
16 simp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> F e. T )
17 12 4 5 ltrncl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. ( Base ` K ) ) -> ( F ` P ) e. ( Base ` K ) )
18 15 16 14 17 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) e. ( Base ` K ) )
19 12 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( F ` P ) e. ( Base ` K ) ) -> ( P .\/ ( F ` P ) ) e. ( Base ` K ) )
20 10 14 18 19 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( F ` P ) ) e. ( Base ` K ) )
21 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> W e. H )
22 12 4 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
23 21 22 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> W e. ( Base ` K ) )
24 12 1 2 latlej2 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( F ` P ) e. ( Base ` K ) ) -> ( F ` P ) .<_ ( P .\/ ( F ` P ) ) )
25 10 14 18 24 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) .<_ ( P .\/ ( F ` P ) ) )
26 eqid 
 |-  ( meet ` K ) = ( meet ` K )
27 12 1 2 26 3 atmod2i1 
 |-  ( ( K e. HL /\ ( ( F ` P ) e. A /\ ( P .\/ ( F ` P ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ ( F ` P ) .<_ ( P .\/ ( F ` P ) ) ) -> ( ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) .\/ ( F ` P ) ) = ( ( P .\/ ( F ` P ) ) ( meet ` K ) ( W .\/ ( F ` P ) ) ) )
28 7 9 20 23 25 27 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) .\/ ( F ` P ) ) = ( ( P .\/ ( F ` P ) ) ( meet ` K ) ( W .\/ ( F ` P ) ) ) )
29 1 3 4 5 ltrnel 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) )
30 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
31 1 2 30 3 4 lhpjat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) ) -> ( W .\/ ( F ` P ) ) = ( 1. ` K ) )
32 7 21 29 31 syl21anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( W .\/ ( F ` P ) ) = ( 1. ` K ) )
33 32 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( P .\/ ( F ` P ) ) ( meet ` K ) ( W .\/ ( F ` P ) ) ) = ( ( P .\/ ( F ` P ) ) ( meet ` K ) ( 1. ` K ) ) )
34 hlol 
 |-  ( K e. HL -> K e. OL )
35 7 34 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> K e. OL )
36 12 26 30 olm11 
 |-  ( ( K e. OL /\ ( P .\/ ( F ` P ) ) e. ( Base ` K ) ) -> ( ( P .\/ ( F ` P ) ) ( meet ` K ) ( 1. ` K ) ) = ( P .\/ ( F ` P ) ) )
37 35 20 36 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( P .\/ ( F ` P ) ) ( meet ` K ) ( 1. ` K ) ) = ( P .\/ ( F ` P ) ) )
38 28 33 37 3eqtrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( F ` P ) ) = ( ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) .\/ ( F ` P ) ) )
39 1 2 26 3 4 5 6 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) )
40 39 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( R ` F ) .\/ ( F ` P ) ) = ( ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) .\/ ( F ` P ) ) )
41 12 4 5 6 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
42 15 16 41 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) e. ( Base ` K ) )
43 12 2 latjcom 
 |-  ( ( K e. Lat /\ ( R ` F ) e. ( Base ` K ) /\ ( F ` P ) e. ( Base ` K ) ) -> ( ( R ` F ) .\/ ( F ` P ) ) = ( ( F ` P ) .\/ ( R ` F ) ) )
44 10 42 18 43 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( R ` F ) .\/ ( F ` P ) ) = ( ( F ` P ) .\/ ( R ` F ) ) )
45 38 40 44 3eqtr2rd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` P ) .\/ ( R ` F ) ) = ( P .\/ ( F ` P ) ) )