Metamath Proof Explorer

Theorem trljat1

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. TODO: shorten with atmod3i1 ? (Contributed by NM, 22-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 trljat1 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( 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 eqid 
 |-  ( meet ` K ) = ( meet ` K )
8 1 2 7 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 ) )
9 8 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( R ` F ) .\/ P ) = ( ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) .\/ P ) )
10 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> K e. HL )
11 10 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> K e. Lat )
12 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> P e. A )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
15 12 14 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> P e. ( Base ` K ) )
16 13 4 5 6 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
17 16 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) e. ( Base ` K ) )
18 13 2 latjcom 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( R ` F ) e. ( Base ` K ) ) -> ( P .\/ ( R ` F ) ) = ( ( R ` F ) .\/ P ) )
19 11 15 17 18 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` F ) ) = ( ( R ` F ) .\/ P ) )
20 13 4 5 ltrncl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. ( Base ` K ) ) -> ( F ` P ) e. ( Base ` K ) )
21 15 20 syld3an3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) e. ( Base ` K ) )
22 13 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( F ` P ) e. ( Base ` K ) ) -> ( P .\/ ( F ` P ) ) e. ( Base ` K ) )
23 11 15 21 22 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( F ` P ) ) e. ( Base ` K ) )
24 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> W e. H )
25 13 4 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
26 24 25 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> W e. ( Base ` K ) )
27 13 1 2 latlej1 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( F ` P ) e. ( Base ` K ) ) -> P .<_ ( P .\/ ( F ` P ) ) )
28 11 15 21 27 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> P .<_ ( P .\/ ( F ` P ) ) )
29 13 1 2 7 3 atmod2i1 
 |-  ( ( K e. HL /\ ( P e. A /\ ( P .\/ ( F ` P ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ P .<_ ( P .\/ ( F ` P ) ) ) -> ( ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) .\/ P ) = ( ( P .\/ ( F ` P ) ) ( meet ` K ) ( W .\/ P ) ) )
30 10 12 23 26 28 29 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) .\/ P ) = ( ( P .\/ ( F ` P ) ) ( meet ` K ) ( W .\/ P ) ) )
31 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
32 1 2 31 3 4 lhpjat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( W .\/ P ) = ( 1. ` K ) )
33 32 3adant2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( W .\/ P ) = ( 1. ` K ) )
34 33 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( P .\/ ( F ` P ) ) ( meet ` K ) ( W .\/ P ) ) = ( ( P .\/ ( F ` P ) ) ( meet ` K ) ( 1. ` K ) ) )
35 hlol 
 |-  ( K e. HL -> K e. OL )
36 10 35 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> K e. OL )
37 13 7 31 olm11 
 |-  ( ( K e. OL /\ ( P .\/ ( F ` P ) ) e. ( Base ` K ) ) -> ( ( P .\/ ( F ` P ) ) ( meet ` K ) ( 1. ` K ) ) = ( P .\/ ( F ` P ) ) )
38 36 23 37 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 ) ) )
39 30 34 38 3eqtrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( F ` P ) ) = ( ( ( P .\/ ( F ` P ) ) ( meet ` K ) W ) .\/ P ) )
40 9 19 39 3eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` F ) ) = ( P .\/ ( F ` P ) ) )