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Theorem trlator0 35997
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 2654 . . . 4  |-  ( ( R `  F )  =/=  .0.  <->  -.  ( R `  F )  =  .0.  )
2 eqid 2457 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
3 trl0a.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
4 trl0a.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
52, 3, 4lhpexnle 35831 . . . . . . 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 759 . . . . . . 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 760 . . . . . . 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 755 . . . . . . . 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 755 . . . . . . . . . . 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 35996 . . . . . . . . . . 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 1232 . . . . . . . . . 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 2681 . . . . . . . 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 35995 . . . . . . 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 1232 . . . . . 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 2953 . . . . 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 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   class class class wbr 4456   ` cfv 5594   lecple 14718   0.cp0 15793   Atomscatm 35089   HLchlt 35176   LHypclh 35809   LTrncltrn 35926   trLctrl 35984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985
This theorem is referenced by:  trlatn0  35998  cdlemg31b0a  36522  trlcone  36555  cdlemkfid1N  36748  tendoex  36802  dia2dimlem2  36893  dia2dimlem3  36894
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