Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trlator0 Unicode version

Theorem trlator0 30653
Description: The trace of a lattice translation is an atom or zero. (Contributed by NM, 5-May-2013.)
Hypotheses
Ref Expression
trl0a.z  |-  .0.  =  ( 0. `  K )
trl0a.a  |-  A  =  ( Atoms `  K )
trl0a.h  |-  H  =  ( LHyp `  K
)
trl0a.t  |-  T  =  ( ( LTrn `  K
) `  W )
trl0a.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlator0  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  \/  ( R `  F )  =  .0.  ) )

Proof of Theorem trlator0
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 df-ne 2569 . . . 4  |-  ( ( R `  F )  =/=  .0.  <->  -.  ( R `  F )  =  .0.  )
2 eqid 2404 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
3 trl0a.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
4 trl0a.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
52, 3, 4lhpexnle 30488 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p ( le `  K ) W )
65ad2antrr 707 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  ->  E. p  e.  A  -.  p
( le `  K
) W )
7 simplll 735 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
8 simpr 448 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( p  e.  A  /\  -.  p
( le `  K
) W ) )
9 simpllr 736 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  F  e.  T )
10 simplr 732 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( R `  F )  =/=  .0.  )
117adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  /\  ( R `  F )  =/=  .0.  )  /\  ( p  e.  A  /\  -.  p
( le `  K
) W ) )  /\  ( F `  p )  =  p )  ->  ( K  e.  HL  /\  W  e.  H ) )
12 simplr 732 . . . . . . . . . . 11  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  /\  ( R `  F )  =/=  .0.  )  /\  ( p  e.  A  /\  -.  p
( le `  K
) W ) )  /\  ( F `  p )  =  p )  ->  ( p  e.  A  /\  -.  p
( le `  K
) W ) )
139adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  /\  ( R `  F )  =/=  .0.  )  /\  ( p  e.  A  /\  -.  p
( le `  K
) W ) )  /\  ( F `  p )  =  p )  ->  F  e.  T )
14 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  /\  ( R `  F )  =/=  .0.  )  /\  ( p  e.  A  /\  -.  p
( le `  K
) W ) )  /\  ( F `  p )  =  p )  ->  ( F `  p )  =  p )
15 trl0a.z . . . . . . . . . . . 12  |-  .0.  =  ( 0. `  K )
16 trl0a.t . . . . . . . . . . . 12  |-  T  =  ( ( LTrn `  K
) `  W )
17 trl0a.r . . . . . . . . . . . 12  |-  R  =  ( ( trL `  K
) `  W )
182, 15, 3, 4, 16, 17trl0 30652 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  -.  p
( le `  K
) W )  /\  ( F  e.  T  /\  ( F `  p
)  =  p ) )  ->  ( R `  F )  =  .0.  )
1911, 12, 13, 14, 18syl112anc 1188 . . . . . . . . . 10  |-  ( ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  /\  ( R `  F )  =/=  .0.  )  /\  ( p  e.  A  /\  -.  p
( le `  K
) W ) )  /\  ( F `  p )  =  p )  ->  ( R `  F )  =  .0.  )
2019ex 424 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( ( F `  p )  =  p  ->  ( R `
 F )  =  .0.  ) )
2120necon3d 2605 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( ( R `  F )  =/=  .0.  ->  ( F `  p )  =/=  p
) )
2210, 21mpd 15 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( F `  p )  =/=  p
)
232, 3, 4, 16, 17trlat 30651 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  -.  p
( le `  K
) W )  /\  ( F  e.  T  /\  ( F `  p
)  =/=  p ) )  ->  ( R `  F )  e.  A
)
247, 8, 9, 22, 23syl112anc 1188 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( R `  F )  e.  A
)
256, 24rexlimddv 2794 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  ->  ( R `  F )  e.  A )
2625ex 424 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  =/=  .0.  ->  ( R `  F )  e.  A
) )
271, 26syl5bir 210 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( -.  ( R `  F )  =  .0.  ->  ( R `  F )  e.  A ) )
2827orrd 368 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  =  .0.  \/  ( R `
 F )  e.  A ) )
2928orcomd 378 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  \/  ( R `  F )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413   lecple 13491   0.cp0 14421   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
This theorem is referenced by:  trlatn0  30654  cdlemg31b0a  31177  trlcone  31210  cdlemkfid1N  31403  tendoex  31457  dia2dimlem2  31548  dia2dimlem3  31549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641
  Copyright terms: Public domain W3C validator