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Theorem trlator0 31030
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 2603 . . . 4  |-  ( ( R `  F )  =/=  .0.  <->  -.  ( R `  F )  =  .0.  )
2 eqid 2438 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
3 trl0a.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
4 trl0a.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
52, 3, 4lhpexnle 30865 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p ( le `  K ) W )
65ad2antrr 708 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  ->  E. p  e.  A  -.  p
( le `  K
) W )
7 simplll 736 . . . . . . 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 449 . . . . . . 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 737 . . . . . . 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 733 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( R `  F )  =/=  .0.  )
117adantr 453 . . . . . . . . . . 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 733 . . . . . . . . . . 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 453 . . . . . . . . . . 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 449 . . . . . . . . . . 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 31029 . . . . . . . . . . 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 1189 . . . . . . . . . 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 425 . . . . . . . . 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 2641 . . . . . . . 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 31028 . . . . . . 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 1189 . . . . . 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 2836 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  ->  ( R `  F )  e.  A )
2625ex 425 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  =/=  .0.  ->  ( R `  F )  e.  A
) )
271, 26syl5bir 211 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( -.  ( R `  F )  =  .0.  ->  ( R `  F )  e.  A ) )
2827orrd 369 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  =  .0.  \/  ( R `
 F )  e.  A ) )
2928orcomd 379 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 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4214   ` cfv 5456   lecple 13538   0.cp0 14468   Atomscatm 30123   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   trLctrl 31017
This theorem is referenced by:  trlatn0  31031  cdlemg31b0a  31554  trlcone  31587  cdlemkfid1N  31780  tendoex  31834  dia2dimlem2  31925  dia2dimlem3  31926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018
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