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Theorem trlator0 33820
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 2613 . . . 4  |-  ( ( R `  F )  =/=  .0.  <->  -.  ( R `  F )  =  .0.  )
2 eqid 2443 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
3 trl0a.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
4 trl0a.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
52, 3, 4lhpexnle 33655 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p ( le `  K ) W )
65ad2antrr 725 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  ->  E. p  e.  A  -.  p
( le `  K
) W )
7 simplll 757 . . . . . . 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 461 . . . . . . 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 758 . . . . . . 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 754 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( R `  F )  =/=  .0.  )
117adantr 465 . . . . . . . . . . 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 754 . . . . . . . . . . 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 465 . . . . . . . . . . 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 461 . . . . . . . . . . 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 33819 . . . . . . . . . . 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 1222 . . . . . . . . . 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 434 . . . . . . . . 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 2651 . . . . . . . 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 33818 . . . . . . 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 1222 . . . . . 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 2850 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  =/=  .0.  )  ->  ( R `  F )  e.  A )
2625ex 434 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  =/=  .0.  ->  ( R `  F )  e.  A
) )
271, 26syl5bir 218 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( -.  ( R `  F )  =  .0.  ->  ( R `  F )  e.  A ) )
2827orrd 378 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  =  .0.  \/  ( R `
 F )  e.  A ) )
2928orcomd 388 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  \/  ( R `  F )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721   class class class wbr 4297   ` cfv 5423   lecple 14250   0.cp0 15212   Atomscatm 32913   HLchlt 33000   LHypclh 33633   LTrncltrn 33750   trLctrl 33807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-clat 15283  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001  df-lhyp 33637  df-laut 33638  df-ldil 33753  df-ltrn 33754  df-trl 33808
This theorem is referenced by:  trlatn0  33821  cdlemg31b0a  34344  trlcone  34377  cdlemkfid1N  34570  tendoex  34624  dia2dimlem2  34715  dia2dimlem3  34716
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