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Metamath Proof Explorer

Theorem trlne

Description: The trace of a lattice translation is not equal to any atom not under the fiducial co-atom W . Part of proof of Lemma C in Crawley p. 112. (Contributed by NM, 25-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 trlne 
|- ( ( ( 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 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> -. P .<_ W )
7 1 3 4 5 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W )
8 7 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) .<_ W )
9 breq1 
 |-  ( P = ( R ` F ) -> ( P .<_ W <-> ( R ` F ) .<_ W ) )
10 8 9 syl5ibrcom 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P = ( R ` F ) -> P .<_ W ) )
11 10 necon3bd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( -. P .<_ W -> P =/= ( R ` F ) ) )
12 6 11 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> P =/= ( R ` F ) )