Metamath Proof Explorer

Theorem ltrnnidn

Description: If a lattice translation is not the identity, then the translation of any atom not under the fiducial co-atom W is different from the atom. Remark above Lemma C in Crawley p. 112. (Contributed by NM, 24-May-2012)

Ref Expression
Hypotheses ltrnnidn.b 
|- B = ( Base ` K )
ltrnnidn.l 
|- .<_ = ( le ` K )
ltrnnidn.a 
|- A = ( Atoms ` K )
ltrnnidn.h 
|- H = ( LHyp ` K )
ltrnnidn.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrnnidn 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) =/= P )

Proof

Step Hyp Ref Expression
1 ltrnnidn.b 
 |-  B = ( Base ` K )
2 ltrnnidn.l 
 |-  .<_ = ( le ` K )
3 ltrnnidn.a 
 |-  A = ( Atoms ` K )
4 ltrnnidn.h 
 |-  H = ( LHyp ` K )
5 ltrnnidn.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> K e. HL )
7 hlatl 
 |-  ( K e. HL -> K e. AtLat )
8 6 7 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> K e. AtLat )
9 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) )
10 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> F e. T )
11 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> F =/= ( _I |` B ) )
12 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
13 1 3 4 5 12 trlnidat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( ( ( trL ` K ) ` W ) ` F ) e. A )
14 9 10 11 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( ( trL ` K ) ` W ) ` F ) e. A )
15 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
16 15 3 atn0 
 |-  ( ( K e. AtLat /\ ( ( ( trL ` K ) ` W ) ` F ) e. A ) -> ( ( ( trL ` K ) ` W ) ` F ) =/= ( 0. ` K ) )
17 8 14 16 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( ( trL ` K ) ` W ) ` F ) =/= ( 0. ` K ) )
18 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> ( K e. HL /\ W e. H ) )
19 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> ( P e. A /\ -. P .<_ W ) )
20 simpl2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> F e. T )
21 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> ( F ` P ) = P )
22 2 15 3 4 5 12 trl0 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( ( ( trL ` K ) ` W ) ` F ) = ( 0. ` K ) )
23 18 19 20 21 22 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F ` P ) = P ) -> ( ( ( trL ` K ) ` W ) ` F ) = ( 0. ` K ) )
24 23 ex 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` P ) = P -> ( ( ( trL ` K ) ` W ) ` F ) = ( 0. ` K ) ) )
25 24 necon3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( ( ( trL ` K ) ` W ) ` F ) =/= ( 0. ` K ) -> ( F ` P ) =/= P ) )
26 17 25 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) =/= P )