Metamath Proof Explorer

Theorem trlat

Description: If an atom differs from its translation, the trace is an atom. Equation above Lemma C in Crawley p. 112. (Contributed by NM, 23-May-2012)

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

Proof

Step Hyp Ref Expression
1 trlat.l 
 |-  .<_ = ( le ` K )
2 trlat.a 
 |-  A = ( Atoms ` K )
3 trlat.h 
 |-  H = ( LHyp ` K )
4 trlat.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 trlat.r 
 |-  R = ( ( trL ` K ) ` W )
6 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( K e. HL /\ W e. H ) )
7 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> F e. T )
8 simp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( P e. A /\ -. P .<_ W ) )
9 eqid 
 |-  ( join ` K ) = ( join ` K )
10 eqid 
 |-  ( meet ` K ) = ( meet ` K )
11 1 9 10 2 3 4 5 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) )
12 6 7 8 11 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) = ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) )
13 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> P e. A )
14 1 2 3 4 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A )
15 6 7 13 14 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( F ` P ) e. A )
16 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( F ` P ) =/= P )
17 16 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> P =/= ( F ` P ) )
18 1 9 10 2 3 lhpat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) e. A /\ P =/= ( F ` P ) ) ) -> ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) e. A )
19 6 8 15 17 18 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) e. A )
20 12 19 eqeltrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A )