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Theorem trlatn0 30361
Description: The trace of a lattice translation is an atom iff it is nonzero. (Contributed by NM, 14-Jun-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
trlatn0  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  <->  ( R `  F )  =/=  .0.  ) )

Proof of Theorem trlatn0
StepHypRef Expression
1 hlatl 29550 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
21ad3antrrr 710 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  e.  A )  ->  K  e.  AtLat )
3 trl0a.z . . . . 5  |-  .0.  =  ( 0. `  K )
4 trl0a.a . . . . 5  |-  A  =  ( Atoms `  K )
53, 4atn0 29498 . . . 4  |-  ( ( K  e.  AtLat  /\  ( R `  F )  e.  A )  ->  ( R `  F )  =/=  .0.  )
62, 5sylancom 648 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  ( R `  F )  e.  A )  ->  ( R `  F )  =/=  .0.  )
76ex 423 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  ->  ( R `
 F )  =/= 
.0.  ) )
8 trl0a.h . . . . 5  |-  H  =  ( LHyp `  K
)
9 trl0a.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
10 trl0a.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
113, 4, 8, 9, 10trlator0 30360 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  \/  ( R `  F )  =  .0.  ) )
1211ord 366 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( -.  ( R `  F )  e.  A  ->  ( R `  F )  =  .0.  ) )
1312necon1ad 2513 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  =/=  .0.  ->  ( R `  F )  e.  A
) )
147, 13impbid 183 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:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255   0.cp0 14143   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  trlid0b  30367  cdlemg12e  30836  trlcoat  30912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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