Metamath Proof Explorer

Theorem trlnle

Description: The atom not under the fiducial co-atom W is not less than the trace of a lattice translation. Part of proof of Lemma C in Crawley p. 112. (Contributed by NM, 26-May-2012)

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

Proof

Step Hyp Ref Expression
1 trlne.l 
 |-  .<_ = ( le ` K )
2 trlne.a 
 |-  A = ( Atoms ` K )
3 trlne.h 
 |-  H = ( LHyp ` K )
4 trlne.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 trlne.r 
 |-  R = ( ( trL ` K ) ` W )
6 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> K e. HL )
7 hlatl 
 |-  ( K e. HL -> K e. AtLat )
8 6 7 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> K e. AtLat )
9 simpl3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> P e. A )
10 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
11 1 10 2 atnle0 
 |-  ( ( K e. AtLat /\ P e. A ) -> -. P .<_ ( 0. ` K ) )
12 8 9 11 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> -. P .<_ ( 0. ` K ) )
13 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> ( K e. HL /\ W e. H ) )
14 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> ( P e. A /\ -. P .<_ W ) )
15 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> F e. T )
16 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> ( F ` P ) = P )
17 1 10 2 3 4 5 trl0 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = ( 0. ` K ) )
18 13 14 15 16 17 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> ( R ` F ) = ( 0. ` K ) )
19 18 breq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> ( P .<_ ( R ` F ) <-> P .<_ ( 0. ` K ) ) )
20 12 19 mtbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> -. P .<_ ( R ` F ) )
21 1 2 3 4 5 trlne 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> P =/= ( R ` F ) )
22 21 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) =/= P ) -> P =/= ( R ` F ) )
23 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) =/= P ) -> K e. HL )
24 23 7 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) =/= P ) -> K e. AtLat )
25 simpl3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) =/= P ) -> P e. A )
26 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) =/= P ) -> ( K e. HL /\ W e. H ) )
27 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) =/= P ) -> ( P e. A /\ -. P .<_ W ) )
28 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) =/= P ) -> F e. T )
29 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) =/= P ) -> ( F ` P ) =/= P )
30 1 2 3 4 5 trlat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A )
31 26 27 28 29 30 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) e. A )
32 1 2 atncmp 
 |-  ( ( K e. AtLat /\ P e. A /\ ( R ` F ) e. A ) -> ( -. P .<_ ( R ` F ) <-> P =/= ( R ` F ) ) )
33 24 25 31 32 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) =/= P ) -> ( -. P .<_ ( R ` F ) <-> P =/= ( R ` F ) ) )
34 22 33 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) =/= P ) -> -. P .<_ ( R ` F ) )
35 20 34 pm2.61dane 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> -. P .<_ ( R ` F ) )