Metamath Proof Explorer

Theorem trlval3

Description: The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (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 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 ) ) ) )

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 simpl1 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( K e. HL /\ W e. H ) )
9 simpl31 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( P e. A /\ -. P .<_ W ) )
10 simpl2 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> F e. T )
11 simpr 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( F ` P ) = P )
12 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
13 1 12 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 ) )
14 8 9 10 11 13 syl112anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( R ` F ) = ( 0. ` K ) )
15 simpl33 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) )
16 simpl1l 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> K e. HL )
17 hlatl 
 |-  ( K e. HL -> K e. AtLat )
18 16 17 syl 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> K e. AtLat )
19 11 oveq2d 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( P .\/ ( F ` P ) ) = ( P .\/ P ) )
20 simp31l 
 |-  ( ( ( 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 ) ) ) ) -> P e. A )
21 20 adantr 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> P e. A )
22 2 4 hlatjidm 
 |-  ( ( K e. HL /\ P e. A ) -> ( P .\/ P ) = P )
23 16 21 22 syl2anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( P .\/ P ) = P )
24 19 23 eqtrd 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( P .\/ ( F ` P ) ) = P )
25 24 21 eqeltrd 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( P .\/ ( F ` P ) ) e. A )
26 simp1 
 |-  ( ( ( 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 ) ) ) ) -> ( K e. HL /\ W e. H ) )
27 simp2 
 |-  ( ( ( 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 ) ) ) ) -> F e. T )
28 simp31 
 |-  ( ( ( 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 ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
29 simp32 
 |-  ( ( ( 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 ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
30 1 4 5 6 ltrn2ateq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( ( F ` P ) = P <-> ( F ` Q ) = Q ) )
31 26 27 28 29 30 syl13anc 
 |-  ( ( ( 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 ) ) ) ) -> ( ( F ` P ) = P <-> ( F ` Q ) = Q ) )
32 31 biimpa 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( F ` Q ) = Q )
33 32 oveq2d 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( F ` Q ) ) = ( Q .\/ Q ) )
34 simp32l 
 |-  ( ( ( 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 ) ) ) ) -> Q e. A )
35 34 adantr 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> Q e. A )
36 2 4 hlatjidm 
 |-  ( ( K e. HL /\ Q e. A ) -> ( Q .\/ Q ) = Q )
37 16 35 36 syl2anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( Q .\/ Q ) = Q )
38 33 37 eqtrd 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( F ` Q ) ) = Q )
39 38 35 eqeltrd 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( Q .\/ ( F ` Q ) ) e. A )
40 3 12 4 atnem0 
 |-  ( ( K e. AtLat /\ ( P .\/ ( F ` P ) ) e. A /\ ( Q .\/ ( F ` Q ) ) e. A ) -> ( ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) <-> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) )
41 18 25 39 40 syl3anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) <-> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) )
42 15 41 mpbid 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) )
43 14 42 eqtr4d 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) = P ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) )
44 simpl1 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( K e. HL /\ W e. H ) )
45 simpl2 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> F e. T )
46 simpl31 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( P e. A /\ -. P .<_ W ) )
47 1 2 3 4 5 6 7 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ W ) )
48 44 45 46 47 syl3anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ W ) )
49 simpl1l 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> K e. HL )
50 49 hllatd 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> K e. Lat )
51 20 adantr 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> P e. A )
52 1 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A )
53 44 45 51 52 syl3anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( F ` P ) e. A )
54 eqid 
 |-  ( Base ` K ) = ( Base ` K )
55 54 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ ( F ` P ) e. A ) -> ( P .\/ ( F ` P ) ) e. ( Base ` K ) )
56 49 51 53 55 syl3anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( P .\/ ( F ` P ) ) e. ( Base ` K ) )
57 simpl1r 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> W e. H )
58 54 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
59 57 58 syl 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> W e. ( Base ` K ) )
60 54 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ ( F ` P ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) .<_ ( P .\/ ( F ` P ) ) )
61 50 56 59 60 syl3anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) .<_ ( P .\/ ( F ` P ) ) )
62 48 61 eqbrtrd 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) .<_ ( P .\/ ( F ` P ) ) )
63 simpl32 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( Q e. A /\ -. Q .<_ W ) )
64 1 2 3 4 5 6 7 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( R ` F ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) )
65 44 45 63 64 syl3anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) )
66 34 adantr 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> Q e. A )
67 1 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ Q e. A ) -> ( F ` Q ) e. A )
68 44 45 66 67 syl3anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( F ` Q ) e. A )
69 54 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ ( F ` Q ) e. A ) -> ( Q .\/ ( F ` Q ) ) e. ( Base ` K ) )
70 49 66 68 69 syl3anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( Q .\/ ( F ` Q ) ) e. ( Base ` K ) )
71 54 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( Q .\/ ( F ` Q ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( Q .\/ ( F ` Q ) ) ./\ W ) .<_ ( Q .\/ ( F ` Q ) ) )
72 50 70 59 71 syl3anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( Q .\/ ( F ` Q ) ) ./\ W ) .<_ ( Q .\/ ( F ` Q ) ) )
73 65 72 eqbrtrd 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) .<_ ( Q .\/ ( F ` Q ) ) )
74 54 5 6 7 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
75 44 45 74 syl2anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) e. ( Base ` K ) )
76 54 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( ( R ` F ) e. ( Base ` K ) /\ ( P .\/ ( F ` P ) ) e. ( Base ` K ) /\ ( Q .\/ ( F ` Q ) ) e. ( Base ` K ) ) ) -> ( ( ( R ` F ) .<_ ( P .\/ ( F ` P ) ) /\ ( R ` F ) .<_ ( Q .\/ ( F ` Q ) ) ) <-> ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) )
77 50 75 56 70 76 syl13anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( ( R ` F ) .<_ ( P .\/ ( F ` P ) ) /\ ( R ` F ) .<_ ( Q .\/ ( F ` Q ) ) ) <-> ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) )
78 62 73 77 mpbi2and 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) )
79 49 17 syl 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> K e. AtLat )
80 simpr 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( F ` P ) =/= P )
81 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 )
82 44 46 45 80 81 syl112anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) e. A )
83 54 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ ( F ` P ) ) e. ( Base ` K ) /\ ( Q .\/ ( F ` Q ) ) e. ( Base ` K ) ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. ( Base ` K ) )
84 50 56 70 83 syl3anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. ( Base ` K ) )
85 54 1 12 4 atlen0 
 |-  ( ( ( K e. AtLat /\ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. ( Base ` K ) /\ ( R ` F ) e. A ) /\ ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) =/= ( 0. ` K ) )
86 79 84 82 78 85 syl31anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) =/= ( 0. ` K ) )
87 86 neneqd 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> -. ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) )
88 simpl33 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) )
89 2 3 12 4 2atmat0 
 |-  ( ( ( K e. HL /\ P e. A /\ ( F ` P ) e. A ) /\ ( Q e. A /\ ( F ` Q ) e. A /\ ( P .\/ ( F ` P ) ) =/= ( Q .\/ ( F ` Q ) ) ) ) -> ( ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. A \/ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) )
90 49 51 53 66 68 88 89 syl33anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. A \/ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) )
91 90 ord 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( -. ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. A -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) = ( 0. ` K ) ) )
92 87 91 mt3d 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. A )
93 1 4 atcmp 
 |-  ( ( K e. AtLat /\ ( R ` F ) e. A /\ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) e. A ) -> ( ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) <-> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) )
94 79 82 92 93 syl3anc 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( ( R ` F ) .<_ ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) <-> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) ) )
95 78 94 mpbid 
 |-  ( ( ( ( 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 ) ) ) ) /\ ( F ` P ) =/= P ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ ( Q .\/ ( F ` Q ) ) ) )
96 43 95 pm2.61dane 
 |-  ( ( ( 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 ) ) ) )