Metamath Proof Explorer

Theorem trlval2

Description: The value of the trace of a lattice translation, given any atom P not under the fiducial co-atom W . Note: this requires only the weaker assumption K e. Lat ; we use K e. HL for convenience. (Contributed by NM, 20-May-2012)

Ref Expression
Hypotheses trlval2.l 
|- .<_ = ( le ` K )
trlval2.j 
|- .\/ = ( join ` K )
trlval2.m 
|- ./\ = ( meet ` K )
trlval2.a 
|- A = ( Atoms ` K )
trlval2.h 
|- H = ( LHyp ` K )
trlval2.t 
|- T = ( ( LTrn ` K ) ` W )
trlval2.r 
|- R = ( ( trL ` K ) ` W )
Assertion trlval2 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ W ) )

Proof

Step Hyp Ref Expression
1 trlval2.l 
 |-  .<_ = ( le ` K )
2 trlval2.j 
 |-  .\/ = ( join ` K )
3 trlval2.m 
 |-  ./\ = ( meet ` K )
4 trlval2.a 
 |-  A = ( Atoms ` K )
5 trlval2.h 
 |-  H = ( LHyp ` K )
6 trlval2.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 trlval2.r 
 |-  R = ( ( trL ` K ) ` W )
8 hllat 
 |-  ( K e. HL -> K e. Lat )
9 8 anim1i 
 |-  ( ( K e. HL /\ W e. H ) -> ( K e. Lat /\ W e. H ) )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 1 2 3 4 5 6 7 trlval 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T ) -> ( R ` F ) = ( iota_ x e. ( Base ` K ) A. q e. A ( -. q .<_ W -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) )
12 11 3adant3 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( iota_ x e. ( Base ` K ) A. q e. A ( -. q .<_ W -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) )
13 simp1l 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> K e. Lat )
14 simp3l 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> P e. A )
15 10 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
16 14 15 syl 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> P e. ( Base ` K ) )
17 10 5 6 ltrncl 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ P e. ( Base ` K ) ) -> ( F ` P ) e. ( Base ` K ) )
18 16 17 syld3an3 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) e. ( Base ` K ) )
19 10 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( F ` P ) e. ( Base ` K ) ) -> ( P .\/ ( F ` P ) ) e. ( Base ` K ) )
20 13 16 18 19 syl3anc 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( F ` P ) ) e. ( Base ` K ) )
21 simp1r 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> W e. H )
22 10 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
23 21 22 syl 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> W e. ( Base ` K ) )
24 10 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ ( F ` P ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) e. ( Base ` K ) )
25 13 20 23 24 syl3anc 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) e. ( Base ` K ) )
26 simpl3l 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ x e. ( Base ` K ) ) -> P e. A )
27 simpl3r 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ x e. ( Base ` K ) ) -> -. P .<_ W )
28 breq1 
 |-  ( q = P -> ( q .<_ W <-> P .<_ W ) )
29 28 notbid 
 |-  ( q = P -> ( -. q .<_ W <-> -. P .<_ W ) )
30 id 
 |-  ( q = P -> q = P )
31 fveq2 
 |-  ( q = P -> ( F ` q ) = ( F ` P ) )
32 30 31 oveq12d 
 |-  ( q = P -> ( q .\/ ( F ` q ) ) = ( P .\/ ( F ` P ) ) )
33 32 oveq1d 
 |-  ( q = P -> ( ( q .\/ ( F ` q ) ) ./\ W ) = ( ( P .\/ ( F ` P ) ) ./\ W ) )
34 33 eqeq2d 
 |-  ( q = P -> ( x = ( ( q .\/ ( F ` q ) ) ./\ W ) <-> x = ( ( P .\/ ( F ` P ) ) ./\ W ) ) )
35 29 34 imbi12d 
 |-  ( q = P -> ( ( -. q .<_ W -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) <-> ( -. P .<_ W -> x = ( ( P .\/ ( F ` P ) ) ./\ W ) ) ) )
36 35 rspcv 
 |-  ( P e. A -> ( A. q e. A ( -. q .<_ W -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) -> ( -. P .<_ W -> x = ( ( P .\/ ( F ` P ) ) ./\ W ) ) ) )
37 36 com23 
 |-  ( P e. A -> ( -. P .<_ W -> ( A. q e. A ( -. q .<_ W -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) -> x = ( ( P .\/ ( F ` P ) ) ./\ W ) ) ) )
38 26 27 37 sylc 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ x e. ( Base ` K ) ) -> ( A. q e. A ( -. q .<_ W -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) -> x = ( ( P .\/ ( F ` P ) ) ./\ W ) ) )
39 simp11 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ -. q .<_ W /\ q e. A ) -> ( K e. Lat /\ W e. H ) )
40 simp12 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ -. q .<_ W /\ q e. A ) -> F e. T )
41 simp13l 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ -. q .<_ W /\ q e. A ) -> P e. A )
42 simp13r 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ -. q .<_ W /\ q e. A ) -> -. P .<_ W )
43 simp3 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ -. q .<_ W /\ q e. A ) -> q e. A )
44 simp2 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ -. q .<_ W /\ q e. A ) -> -. q .<_ W )
45 1 2 3 4 5 6 ltrnu 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( q e. A /\ -. q .<_ W ) ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) )
46 39 40 41 42 43 44 45 syl222anc 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ -. q .<_ W /\ q e. A ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) )
47 eqeq2 
 |-  ( ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) -> ( x = ( ( P .\/ ( F ` P ) ) ./\ W ) <-> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) )
48 47 biimpd 
 |-  ( ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) -> ( x = ( ( P .\/ ( F ` P ) ) ./\ W ) -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) )
49 46 48 syl 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ -. q .<_ W /\ q e. A ) -> ( x = ( ( P .\/ ( F ` P ) ) ./\ W ) -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) )
50 49 3exp 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( -. q .<_ W -> ( q e. A -> ( x = ( ( P .\/ ( F ` P ) ) ./\ W ) -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) ) )
51 50 com24 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( x = ( ( P .\/ ( F ` P ) ) ./\ W ) -> ( q e. A -> ( -. q .<_ W -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) ) )
52 51 ralrimdv 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( x = ( ( P .\/ ( F ` P ) ) ./\ W ) -> A. q e. A ( -. q .<_ W -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) )
53 52 adantr 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ x e. ( Base ` K ) ) -> ( x = ( ( P .\/ ( F ` P ) ) ./\ W ) -> A. q e. A ( -. q .<_ W -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) )
54 38 53 impbid 
 |-  ( ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ x e. ( Base ` K ) ) -> ( A. q e. A ( -. q .<_ W -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) <-> x = ( ( P .\/ ( F ` P ) ) ./\ W ) ) )
55 25 54 riota5 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( iota_ x e. ( Base ` K ) A. q e. A ( -. q .<_ W -> x = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) = ( ( P .\/ ( F ` P ) ) ./\ W ) )
56 12 55 eqtrd 
 |-  ( ( ( K e. Lat /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ W ) )
57 9 56 syl3an1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P .\/ ( F ` P ) ) ./\ W ) )