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Metamath Proof Explorer

Theorem trlval4

Description: The value of the trace of a lattice translation in terms of 2 atoms. (Contributed by NM, 3-May-2013)

Ref Expression
Hypotheses trlval3.l 
|- .<_ = ( le ` K )
trlval3.j 
|- .\/ = ( join ` K )
trlval3.m 
|- ./\ = ( meet ` K )
trlval3.a 
|- A = ( Atoms ` K )
trlval3.h 
|- H = ( LHyp ` K )
trlval3.t 
|- T = ( ( LTrn ` K ) ` W )
trlval3.r 
|- R = ( ( trL ` K ) ` W )
Assertion trlval4 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) )

Proof

Step Hyp Ref Expression
1 trlval3.l 
 |-  .<_ = ( le ` K )
2 trlval3.j 
 |-  .\/ = ( join ` K )
3 trlval3.m 
 |-  ./\ = ( meet ` K )
4 trlval3.a 
 |-  A = ( Atoms ` K )
5 trlval3.h 
 |-  H = ( LHyp ` K )
6 trlval3.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 trlval3.r 
 |-  R = ( ( trL ` K ) ` W )
8 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
9 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> F e. T )
10 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
12 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> -. ( R ` F ) .<_ ( P .\/ Q ) )
13 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> K e. HL )
14 simp23l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> Q e. A )
15 14 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> Q e. A )
16 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( K e. HL /\ W e. H ) )
17 simpl21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> F e. T )
18 1 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ Q e. A ) -> ( F ` Q ) e. A )
19 16 17 15 18 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( F ` Q ) e. A )
20 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ Q e. A /\ ( F ` Q ) e. A ) -> Q .<_ ( Q .\/ ( F ` Q ) ) )
21 13 15 19 20 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> Q .<_ ( Q .\/ ( F ` Q ) ) )
22 simpl22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
23 1 2 4 5 6 7 trljat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` F ) ) = ( P .\/ ( F ` P ) ) )
24 16 17 22 23 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( P .\/ ( R ` F ) ) = ( P .\/ ( F ` P ) ) )
25 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) )
26 24 25 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( P .\/ ( R ` F ) ) = ( Q .\/ ( F ` Q ) ) )
27 21 26 breqtrrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> Q .<_ ( P .\/ ( R ` F ) ) )
28 simpl3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> -. ( R ` F ) .<_ ( P .\/ Q ) )
29 simpll1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( K e. HL /\ W e. H ) )
30 22 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( P e. A /\ -. P .<_ W ) )
31 17 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> F e. T )
32 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( F ` P ) = P )
33 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
34 1 33 4 5 6 7 trl0 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = ( 0. ` K ) )
35 29 30 31 32 34 syl112anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( R ` F ) = ( 0. ` K ) )
36 hlatl 
 |-  ( K e. HL -> K e. AtLat )
37 13 36 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> K e. AtLat )
38 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> P e. A )
39 38 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> P e. A )
40 eqid 
 |-  ( Base ` K ) = ( Base ` K )
41 40 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
42 13 39 15 41 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
43 40 1 33 atl0le 
 |-  ( ( K e. AtLat /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( 0. ` K ) .<_ ( P .\/ Q ) )
44 37 42 43 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( 0. ` K ) .<_ ( P .\/ Q ) )
45 44 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( 0. ` K ) .<_ ( P .\/ Q ) )
46 35 45 eqbrtrd 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) /\ ( F ` P ) = P ) -> ( R ` F ) .<_ ( P .\/ Q ) )
47 46 ex 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( ( F ` P ) = P -> ( R ` F ) .<_ ( P .\/ Q ) ) )
48 47 necon3bd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( -. ( R ` F ) .<_ ( P .\/ Q ) -> ( F ` P ) =/= P ) )
49 28 48 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( F ` P ) =/= P )
50 1 4 5 6 7 trlat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A )
51 16 22 17 49 50 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( R ` F ) e. A )
52 simpl3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> P =/= Q )
53 52 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> Q =/= P )
54 1 2 4 hlatexch1 
 |-  ( ( K e. HL /\ ( Q e. A /\ ( R ` F ) e. A /\ P e. A ) /\ Q =/= P ) -> ( Q .<_ ( P .\/ ( R ` F ) ) -> ( R ` F ) .<_ ( P .\/ Q ) ) )
55 13 15 51 39 53 54 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( Q .<_ ( P .\/ ( R ` F ) ) -> ( R ` F ) .<_ ( P .\/ Q ) ) )
56 27 55 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) ) -> ( R ` F ) .<_ ( P .\/ Q ) )
57 56 ex 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` P ) ) = ( Q .\/ ( F ` Q ) ) -> ( R ` F ) .<_ ( P .\/ Q ) ) )
58 57 necon3bd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( -. ( R ` F ) .<_ ( P .\/ Q ) -> ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) )
59 12 58 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) )
60 1 2 3 4 5 6 7 trlval3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) )
61 8 9 10 11 59 60 syl113anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) )