Metamath Proof Explorer

Theorem trlnidatb

Description: A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat ? Why do both this and ltrnideq need trlnidat ? (Contributed by NM, 4-Jun-2013)

Ref Expression
Hypotheses trlnidatb.b 
|- B = ( Base ` K )
trlnidatb.a 
|- A = ( Atoms ` K )
trlnidatb.h 
|- H = ( LHyp ` K )
trlnidatb.t 
|- T = ( ( LTrn ` K ) ` W )
trlnidatb.r 
|- R = ( ( trL ` K ) ` W )
Assertion trlnidatb 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F =/= ( _I |` B ) <-> ( R ` F ) e. A ) )

Proof

Step Hyp Ref Expression
1 trlnidatb.b 
 |-  B = ( Base ` K )
2 trlnidatb.a 
 |-  A = ( Atoms ` K )
3 trlnidatb.h 
 |-  H = ( LHyp ` K )
4 trlnidatb.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 trlnidatb.r 
 |-  R = ( ( trL ` K ) ` W )
6 1 2 3 4 5 trlnidat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( R ` F ) e. A )
7 6 3expia 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F =/= ( _I |` B ) -> ( R ` F ) e. A ) )
8 eqid 
 |-  ( le ` K ) = ( le ` K )
9 8 2 3 lhpexnle 
 |-  ( ( K e. HL /\ W e. H ) -> E. p e. A -. p ( le ` K ) W )
10 9 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> E. p e. A -. p ( le ` K ) W )
11 1 8 2 3 4 ltrnideq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( F = ( _I |` B ) <-> ( F ` p ) = p ) )
12 11 3expa 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( F = ( _I |` B ) <-> ( F ` p ) = p ) )
13 simp1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F ` p ) = p ) -> ( K e. HL /\ W e. H ) )
14 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F ` p ) = p ) -> ( p e. A /\ -. p ( le ` K ) W ) )
15 simp1r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F ` p ) = p ) -> F e. T )
16 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F ` p ) = p ) -> ( F ` p ) = p )
17 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
18 8 17 2 3 4 5 trl0 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F e. T /\ ( F ` p ) = p ) ) -> ( R ` F ) = ( 0. ` K ) )
19 13 14 15 16 18 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F ` p ) = p ) -> ( R ` F ) = ( 0. ` K ) )
20 19 3expia 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( ( F ` p ) = p -> ( R ` F ) = ( 0. ` K ) ) )
21 simplll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> K e. HL )
22 hlatl 
 |-  ( K e. HL -> K e. AtLat )
23 17 2 atn0 
 |-  ( ( K e. AtLat /\ ( R ` F ) e. A ) -> ( R ` F ) =/= ( 0. ` K ) )
24 23 ex 
 |-  ( K e. AtLat -> ( ( R ` F ) e. A -> ( R ` F ) =/= ( 0. ` K ) ) )
25 21 22 24 3syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( ( R ` F ) e. A -> ( R ` F ) =/= ( 0. ` K ) ) )
26 25 necon2bd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( ( R ` F ) = ( 0. ` K ) -> -. ( R ` F ) e. A ) )
27 20 26 syld 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( ( F ` p ) = p -> -. ( R ` F ) e. A ) )
28 12 27 sylbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ ( p e. A /\ -. p ( le ` K ) W ) ) -> ( F = ( _I |` B ) -> -. ( R ` F ) e. A ) )
29 10 28 rexlimddv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F = ( _I |` B ) -> -. ( R ` F ) e. A ) )
30 29 necon2ad 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( ( R ` F ) e. A -> F =/= ( _I |` B ) ) )
31 7 30 impbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F =/= ( _I |` B ) <-> ( R ` F ) e. A ) )