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Theorem trlval2 33437
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.)
Hypotheses
Ref Expression
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
Ref Expression
trlval2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
)  ./\  W )
)

Proof of Theorem trlval2
Dummy variables  x  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hllat 32637 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
21anim1i 570 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  Lat  /\  W  e.  H ) )
3 eqid 2429 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
4 trlval2.l . . . . 5  |-  .<_  =  ( le `  K )
5 trlval2.j . . . . 5  |-  .\/  =  ( join `  K )
6 trlval2.m . . . . 5  |-  ./\  =  ( meet `  K )
7 trlval2.a . . . . 5  |-  A  =  ( Atoms `  K )
8 trlval2.h . . . . 5  |-  H  =  ( LHyp `  K
)
9 trlval2.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
10 trlval2.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
113, 4, 5, 6, 7, 8, 9, 10trlval 33436 . . . 4  |-  ( ( ( 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 ) ) ) )
12113adant3 1025 . . 3  |-  ( ( ( 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 1029 . . . . 5  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
14 simp3l 1033 . . . . . . 7  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
153, 7atbase 32563 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1614, 15syl 17 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
173, 8, 9ltrncl 33398 . . . . . . 7  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( F `  P )  e.  (
Base `  K )
)
1816, 17syld3an3 1309 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  e.  (
Base `  K )
)
193, 5latjcl 16248 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  ->  ( P  .\/  ( F `  P ) )  e.  ( Base `  K
) )
2013, 16, 18, 19syl3anc 1264 . . . . 5  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( F `  P
) )  e.  (
Base `  K )
)
21 simp1r 1030 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
223, 8lhpbase 33271 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2321, 22syl 17 . . . . 5  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K )
)
243, 6latmcl 16249 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  ( F `
 P ) )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  ( F `  P ) )  ./\  W )  e.  ( Base `  K ) )
2513, 20, 23, 24syl3anc 1264 . . . 4  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  ( F `  P ) )  ./\  W )  e.  ( Base `  K ) )
26 simpl3l 1060 . . . . . 6  |-  ( ( ( ( K  e. 
Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  x  e.  ( Base `  K
) )  ->  P  e.  A )
27 simpl3r 1061 . . . . . 6  |-  ( ( ( ( K  e. 
Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  x  e.  ( Base `  K
) )  ->  -.  P  .<_  W )
28 breq1 4429 . . . . . . . . . 10  |-  ( q  =  P  ->  (
q  .<_  W  <->  P  .<_  W ) )
2928notbid 295 . . . . . . . . 9  |-  ( q  =  P  ->  ( -.  q  .<_  W  <->  -.  P  .<_  W ) )
30 id 23 . . . . . . . . . . . 12  |-  ( q  =  P  ->  q  =  P )
31 fveq2 5881 . . . . . . . . . . . 12  |-  ( q  =  P  ->  ( F `  q )  =  ( F `  P ) )
3230, 31oveq12d 6323 . . . . . . . . . . 11  |-  ( q  =  P  ->  (
q  .\/  ( F `  q ) )  =  ( P  .\/  ( F `  P )
) )
3332oveq1d 6320 . . . . . . . . . 10  |-  ( q  =  P  ->  (
( q  .\/  ( F `  q )
)  ./\  W )  =  ( ( P 
.\/  ( F `  P ) )  ./\  W ) )
3433eqeq2d 2443 . . . . . . . . 9  |-  ( q  =  P  ->  (
x  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W )  <->  x  =  ( ( P  .\/  ( F `  P ) )  ./\  W )
) )
3529, 34imbi12d 321 . . . . . . . 8  |-  ( q  =  P  ->  (
( -.  q  .<_  W  ->  x  =  ( ( q  .\/  ( F `  q )
)  ./\  W )
)  <->  ( -.  P  .<_  W  ->  x  =  ( ( P  .\/  ( F `  P ) )  ./\  W )
) ) )
3635rspcv 3184 . . . . . . 7  |-  ( P  e.  A  ->  ( A. q  e.  A  ( -.  q  .<_  W  ->  x  =  ( ( q  .\/  ( F `  q )
)  ./\  W )
)  ->  ( -.  P  .<_  W  ->  x  =  ( ( P 
.\/  ( F `  P ) )  ./\  W ) ) ) )
3736com23 81 . . . . . 6  |-  ( P  e.  A  ->  ( -.  P  .<_  W  -> 
( A. q  e.  A  ( -.  q  .<_  W  ->  x  =  ( ( q  .\/  ( F `  q ) )  ./\  W )
)  ->  x  =  ( ( P  .\/  ( F `  P ) )  ./\  W )
) ) )
3826, 27, 37sylc 62 . . . . 5  |-  ( ( ( ( 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 1035 . . . . . . . . . . 11  |-  ( ( ( ( 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 1036 . . . . . . . . . . 11  |-  ( ( ( ( K  e. 
Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  -.  q  .<_  W  /\  q  e.  A )  ->  F  e.  T )
41 simp13l 1120 . . . . . . . . . . 11  |-  ( ( ( ( K  e. 
Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  -.  q  .<_  W  /\  q  e.  A )  ->  P  e.  A )
42 simp13r 1121 . . . . . . . . . . 11  |-  ( ( ( ( K  e. 
Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  -.  q  .<_  W  /\  q  e.  A )  ->  -.  P  .<_  W )
43 simp3 1007 . . . . . . . . . . 11  |-  ( ( ( ( K  e. 
Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  -.  q  .<_  W  /\  q  e.  A )  ->  q  e.  A )
44 simp2 1006 . . . . . . . . . . 11  |-  ( ( ( ( K  e. 
Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  -.  q  .<_  W  /\  q  e.  A )  ->  -.  q  .<_  W )
454, 5, 6, 7, 8, 9ltrnu 33394 . . . . . . . . . . 11  |-  ( ( ( ( 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 ) )
4639, 40, 41, 42, 43, 44, 45syl222anc 1280 . . . . . . . . . 10  |-  ( ( ( ( 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 2444 . . . . . . . . . . 11  |-  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W )  ->  ( x  =  ( ( P 
.\/  ( F `  P ) )  ./\  W )  <->  x  =  (
( q  .\/  ( F `  q )
)  ./\  W )
) )
4847biimpd 210 . . . . . . . . . 10  |-  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W )  ->  ( x  =  ( ( P 
.\/  ( F `  P ) )  ./\  W )  ->  x  =  ( ( q  .\/  ( F `  q ) )  ./\  W )
) )
4946, 48syl 17 . . . . . . . . 9  |-  ( ( ( ( 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 ) ) )
50493exp 1204 . . . . . . . 8  |-  ( ( ( 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 ) ) ) ) )
5150com24 90 . . . . . . 7  |-  ( ( ( 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 ) ) ) ) )
5251ralrimdv 2848 . . . . . 6  |-  ( ( ( 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 ) ) ) )
5352adantr 466 . . . . 5  |-  ( ( ( ( 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 )
) ) )
5438, 53impbid 193 . . . 4  |-  ( ( ( ( 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 )
) )
5525, 54riota5 6292 . . 3  |-  ( ( ( 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 )
)
5612, 55eqtrd 2470 . 2  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
)  ./\  W )
)
572, 56syl3an1 1297 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
)  ./\  W )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   class class class wbr 4426   ` cfv 5601   iota_crio 6266  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   meetcmee 16141   Latclat 16242   Atomscatm 32537   HLchlt 32624   LHypclh 33257   LTrncltrn 33374   trLctrl 33432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-lat 16243  df-ats 32541  df-atl 32572  df-cvlat 32596  df-hlat 32625  df-lhyp 33261  df-laut 33262  df-ldil 33377  df-ltrn 33378  df-trl 33433
This theorem is referenced by:  trlcl  33438  trlcnv  33439  trljat1  33440  trljat2  33441  trlat  33443  trl0  33444  trlle  33458  trlval3  33461  trlval5  33463  cdlemd6  33477  cdlemf  33838  cdlemg4a  33883  cdlemg4b1  33884  cdlemg4b2  33885  cdlemg4  33892  cdlemg11b  33917  cdlemg13a  33926  cdlemg13  33927  cdlemg17a  33936  cdlemg17dN  33938  cdlemg17e  33940  cdlemg17f  33941  trlcoabs2N  33997  trlcolem  34001  cdlemg42  34004  cdlemg43  34005  cdlemi1  34093  cdlemk4  34109  cdlemk39  34191  dia2dimlem1  34340  dia2dimlem2  34341  dia2dimlem3  34342  cdlemm10N  34394  cdlemn2  34471  cdlemn10  34482  dihjatcclem3  34696
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