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Theorem trlval2 36304
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 35504 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
21anim1i 566 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  Lat  /\  W  e.  H ) )
3 eqid 2454 . . . . 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 36303 . . . 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 1014 . . 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 1018 . . . . 5  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
14 simp3l 1022 . . . . . . 7  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
153, 7atbase 35430 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1614, 15syl 16 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
173, 8, 9ltrncl 36265 . . . . . . 7  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( F `  P )  e.  (
Base `  K )
)
1816, 17syld3an3 1271 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  e.  (
Base `  K )
)
193, 5latjcl 15883 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  ->  ( P  .\/  ( F `  P ) )  e.  ( Base `  K
) )
2013, 16, 18, 19syl3anc 1226 . . . . 5  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( F `  P
) )  e.  (
Base `  K )
)
21 simp1r 1019 . . . . . 6  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
223, 8lhpbase 36138 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2321, 22syl 16 . . . . 5  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K )
)
243, 6latmcl 15884 . . . . 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 1226 . . . 4  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  ( F `  P ) )  ./\  W )  e.  ( Base `  K ) )
26 simpl3l 1049 . . . . . 6  |-  ( ( ( ( K  e. 
Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  x  e.  ( Base `  K
) )  ->  P  e.  A )
27 simpl3r 1050 . . . . . 6  |-  ( ( ( ( K  e. 
Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  x  e.  ( Base `  K
) )  ->  -.  P  .<_  W )
28 breq1 4442 . . . . . . . . . 10  |-  ( q  =  P  ->  (
q  .<_  W  <->  P  .<_  W ) )
2928notbid 292 . . . . . . . . 9  |-  ( q  =  P  ->  ( -.  q  .<_  W  <->  -.  P  .<_  W ) )
30 id 22 . . . . . . . . . . . 12  |-  ( q  =  P  ->  q  =  P )
31 fveq2 5848 . . . . . . . . . . . 12  |-  ( q  =  P  ->  ( F `  q )  =  ( F `  P ) )
3230, 31oveq12d 6288 . . . . . . . . . . 11  |-  ( q  =  P  ->  (
q  .\/  ( F `  q ) )  =  ( P  .\/  ( F `  P )
) )
3332oveq1d 6285 . . . . . . . . . 10  |-  ( q  =  P  ->  (
( q  .\/  ( F `  q )
)  ./\  W )  =  ( ( P 
.\/  ( F `  P ) )  ./\  W ) )
3433eqeq2d 2468 . . . . . . . . 9  |-  ( q  =  P  ->  (
x  =  ( ( q  .\/  ( F `
 q ) ) 
./\  W )  <->  x  =  ( ( P  .\/  ( F `  P ) )  ./\  W )
) )
3529, 34imbi12d 318 . . . . . . . 8  |-  ( q  =  P  ->  (
( -.  q  .<_  W  ->  x  =  ( ( q  .\/  ( F `  q )
)  ./\  W )
)  <->  ( -.  P  .<_  W  ->  x  =  ( ( P  .\/  ( F `  P ) )  ./\  W )
) ) )
3635rspcv 3203 . . . . . . 7  |-  ( P  e.  A  ->  ( A. q  e.  A  ( -.  q  .<_  W  ->  x  =  ( ( q  .\/  ( F `  q )
)  ./\  W )
)  ->  ( -.  P  .<_  W  ->  x  =  ( ( P 
.\/  ( F `  P ) )  ./\  W ) ) ) )
3736com23 78 . . . . . 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 60 . . . . 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 1024 . . . . . . . . . . 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 1025 . . . . . . . . . . 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 1109 . . . . . . . . . . 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 1110 . . . . . . . . . . 11  |-  ( ( ( ( K  e. 
Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  -.  q  .<_  W  /\  q  e.  A )  ->  -.  P  .<_  W )
43 simp3 996 . . . . . . . . . . 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 995 . . . . . . . . . . 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 36261 . . . . . . . . . . 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 1242 . . . . . . . . . 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 2469 . . . . . . . . . . 11  |-  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W )  ->  ( x  =  ( ( P 
.\/  ( F `  P ) )  ./\  W )  <->  x  =  (
( q  .\/  ( F `  q )
)  ./\  W )
) )
4847biimpd 207 . . . . . . . . . 10  |-  ( ( ( P  .\/  ( F `  P )
)  ./\  W )  =  ( ( q 
.\/  ( F `  q ) )  ./\  W )  ->  ( x  =  ( ( P 
.\/  ( F `  P ) )  ./\  W )  ->  x  =  ( ( q  .\/  ( F `  q ) )  ./\  W )
) )
4946, 48syl 16 . . . . . . . . 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 1193 . . . . . . . 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 87 . . . . . . 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 2870 . . . . . 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 463 . . . . 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 191 . . . 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 6257 . . 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 2495 . 2  |-  ( ( ( K  e.  Lat  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
)  ./\  W )
)
572, 56syl3an1 1259 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   class class class wbr 4439   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   meetcmee 15776   Latclat 15877   Atomscatm 35404   HLchlt 35491   LHypclh 36124   LTrncltrn 36241   trLctrl 36299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-lat 15878  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-lhyp 36128  df-laut 36129  df-ldil 36244  df-ltrn 36245  df-trl 36300
This theorem is referenced by:  trlcl  36305  trlcnv  36306  trljat1  36307  trljat2  36308  trlat  36310  trl0  36311  trlle  36325  trlval3  36328  trlval5  36330  cdlemd6  36344  cdlemf  36705  cdlemg4a  36750  cdlemg4b1  36751  cdlemg4b2  36752  cdlemg4  36759  cdlemg11b  36784  cdlemg13a  36793  cdlemg13  36794  cdlemg17a  36803  cdlemg17dN  36805  cdlemg17e  36807  cdlemg17f  36808  trlcoabs2N  36864  trlcolem  36868  cdlemg42  36871  cdlemg43  36872  cdlemi1  36960  cdlemk4  36976  cdlemk39  37058  dia2dimlem1  37207  dia2dimlem2  37208  dia2dimlem3  37209  cdlemm10N  37261  cdlemn2  37338  cdlemn10  37349  dihjatcclem3  37563
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