Metamath Proof Explorer

Theorem trl0

Description: If an atom not under the fiducial co-atom W equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012)

Ref Expression
Hypotheses trl0.l 
|- .<_ = ( le ` K )
trl0.z 
|- .0. = ( 0. ` K )
trl0.a 
|- A = ( Atoms ` K )
trl0.h 
|- H = ( LHyp ` K )
trl0.t 
|- T = ( ( LTrn ` K ) ` W )
trl0.r 
|- R = ( ( trL ` K ) ` W )
Assertion trl0 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = .0. )

Proof

Step Hyp Ref Expression
1 trl0.l 
 |-  .<_ = ( le ` K )
2 trl0.z 
 |-  .0. = ( 0. ` K )
3 trl0.a 
 |-  A = ( Atoms ` K )
4 trl0.h 
 |-  H = ( LHyp ` K )
5 trl0.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 trl0.r 
 |-  R = ( ( trL ` K ) ` W )
7 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( K e. HL /\ W e. H ) )
8 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> F e. T )
9 simp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P e. A /\ -. P .<_ W ) )
10 eqid 
 |-  ( join ` K ) = ( join ` K )
11 eqid 
 |-  ( meet ` K ) = ( meet ` K )
12 1 10 11 3 4 5 6 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) )
13 7 8 9 12 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) )
14 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( F ` P ) = P )
15 14 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( join ` K ) ( F ` P ) ) = ( P ( join ` K ) P ) )
16 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> K e. HL )
17 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> P e. A )
18 10 3 hlatjidm 
 |-  ( ( K e. HL /\ P e. A ) -> ( P ( join ` K ) P ) = P )
19 16 17 18 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( join ` K ) P ) = P )
20 15 19 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( join ` K ) ( F ` P ) ) = P )
21 20 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) = ( P ( meet ` K ) W ) )
22 1 11 2 3 4 lhpmat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ( meet ` K ) W ) = .0. )
23 7 9 22 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( meet ` K ) W ) = .0. )
24 13 21 23 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = .0. )