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Metamath Proof Explorer

Theorem trlnidat

Description: The trace of a lattice translation other than the identity is an atom. Remark above Lemma C in Crawley p. 112. (Contributed by NM, 23-May-2012)

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

Proof

Step Hyp Ref Expression
1 trlnidat.b 
 |-  B = ( Base ` K )
2 trlnidat.a 
 |-  A = ( Atoms ` K )
3 trlnidat.h 
 |-  H = ( LHyp ` K )
4 trlnidat.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 trlnidat.r 
 |-  R = ( ( trL ` K ) ` W )
6 eqid 
 |-  ( le ` K ) = ( le ` K )
7 1 6 2 3 4 ltrnnid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> E. p e. A ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) )
8 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> ( K e. HL /\ W e. H ) )
9 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> p e. A )
10 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> -. p ( le ` K ) W )
11 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> F e. T )
12 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> ( F ` p ) =/= p )
13 6 2 3 4 5 trlat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ -. p ( le ` K ) W ) /\ ( F e. T /\ ( F ` p ) =/= p ) ) -> ( R ` F ) e. A )
14 8 9 10 11 12 13 syl122anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) /\ p e. A /\ ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) ) -> ( R ` F ) e. A )
15 14 rexlimdv3a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( E. p e. A ( -. p ( le ` K ) W /\ ( F ` p ) =/= p ) -> ( R ` F ) e. A ) )
16 7 15 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( R ` F ) e. A )